International Mathematics Research Notices

Beliaev, Dmitry; Cammarota, Valentina; Wigman, Igor

doi: 10.1093/imrn/rnx197pmid: N/A

Abstract Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius. 1 Introduction and Main Results 1.1 Random Gaussian functions Studying the Laplace eigenfunctions and their geometry is a classical subject going back to at least XIX century. It is most important to understand the eigenfunctions behaviour in the high energy limit. For a given domain, this is a difficult question, and we only have limited information about it other than in the few cases where the eigenfunctions is explicitly given. For a generic chaotic domain (i.e., where the billiard dynamics is chaotic) it was conjectured by Berry [3] that the high energy functions behave like a random superposition of monochromatic plane waves propagating in different (random) directions, usually referred to as the random plane wave, rigorously defined below. As the comparison between these two is lacking mathematical rigour, one may understand this comparison in different ways. Berry’s conjecture seems to be out of reach by modern analytic techniques; a similar statement for a random linear combination of eigenfunctions with close eigenvalues could be proved though. Namely, for a compact Riemannian manifold |$\mathcal{M}$| we can consider an orthonormal basis of eigenfunctions |$\phi_i$| satisfying |$\Delta \phi_i+t_i^2\phi_i=0$| with |$t_0\le t_1\le \dots$|⁠, and define the band-limited functions $$ f_T=\sum_{T-\sqrt{T}\le t_i\le T} c_i\phi_i $$ where, |$c_i$| are i.i.d. normal random variables. It is known [6, 9, 10, 15] that the local scaling limit of |$f_T$| is the random plane wave. The above conjectures and results show that the random plane wave is a universal object, and motivate their further study; here we are interested in their geometry. As usual, a Gaussian random field could be defined or constructed in two different ways. On one hand we may define it as a concrete random series, an on the other hand we may describe it as uniquely defined in terms of it covariance function via Kolmogorov’s Theorem. As a concrete random series we define the random plane wave with energy |$E=k^2$| to be \begin{equation} \label{eq:RPW definition} \Psi (z)=\Psi (r,\theta )=\operatorname{Re}\sum\limits_{n=-\infty }^{\infty }{{{a}_{n}}}{{J}_{|n|}}(kr){{e}^{in\theta }} \end{equation} (1) in polar coordinates, where |$J_n$| are Bessel functions and |$a_n$| are independent complex Gaussian random variables with variance |$2$|⁠. Since the Bessel functions decay exponentially fast as functions of the order |$n$|⁠, the series (1) is almost surely convergent, absolutely and uniformly on any compact set, and hence the sum is a real analytic function. By the definition (1), |$\Psi$| is a centred Gaussian random field, therefore its law is prescribed by the covariance function \begin{align*} \psi(z,w):=\mathbb{E}[\Psi(z)\cdot \Psi(w)], \end{align*} for |$z,w \in {\mathbb R}^2$|⁠. It is then easy to evaluate |$\psi$| explicitly as \[ \psi(z,w)=J_0(k|z-w|), \] where |$J_{0}$| is the Bessel |$J$| function of order |$0$|⁠. From this representation it follows that |$\Psi$| is stationary (i.e., its law is translation invariant), and isotropic, (i.e., its law is invariant under rotations); by the standard abuse of notation we write |$\psi(z,w)=\psi(z-w)$|⁠. It also follows directly from (1) that the function |$\Psi$| is an a.s. solution of the Helmholtz equation \begin{align} \label{eq:Helmholz} (\Delta + k^2) \Psi = 0, \end{align} (2) that is |$\Psi$| is an a.s. eigenfunction of |$-\Delta$| with eigenvalue |$k^2$|⁠; we are interested in the geometry of random (or “typical”) solutions |$\Psi$| of (2). The geometric properties considered below are related to the nodal lines (i.e., |$\Psi^{-1}(0)$|⁠), nodal domains (i.e., connected components of the complement of the nodal set), as well as the level curves (⁠|$\Psi^{-1}(c)$|⁠), and excursion sets (connected components of |$\{z:\: \Psi(z)>c\}$|⁠). The geometry of these sets is closely related to that of the set of critical points of |$\Psi$|⁠. The critical points and values and their applications appear a lot ([7, 8, 11–14] to mention a few) in the literature on nodal domains of random plane waves and, more generally, smooth Gaussian fields. 1.2 Critical points There are several intriguing questions on the critical points of random fields. From our perspective, of the most important questions are the ones on the distribution of the critical points number, and the corresponding critical values. This general question could be made more concrete in different ways, most basically, evaluating the expected number of critical points inside a given domain; the latter admits a precise answer in a more general scenario. In an analogous case of random spherical harmonics (that converges to |$\Psi$| as a scaling limit), Cammarota, Marinucci and Wigman [4] evaluated the expected number of minima, maxima and saddles whose value falls into a given window, and also determined the order of magnitude of the corresponding variance for a “generic” window, which does not include the total number of critical points. In a subsequent work Cammarota and Wigman [5] resolved this outstanding case by evaluating the variance of the total number of critical points to be of lower order as compared to the generic case. It is important to understand the finer aspects of the structure of critical points. Upon looking at Figure 1 (left), it is evident that the structure of critical points is very “rigid” or “regular”; however it is not entirely clear how to formulate or quantify this statement with mathematical rigour. One can compare this to two other very well known translation invariant processes: in Figure 1 (centre) one may observe the Poisson point process, and Figure 1 (right) shows the corresponding picture for Ginibre point process; both are scaled to have the same intensity as the critical points in Figure 1 (left). Fig. 1. Open in new tabDownload slide Left: critical points of a random plane wave. Center: The Poisson point process which has the same density. Right: a bulk part of the Ginibre ensemble with the same density. Fig. 1. Open in new tabDownload slide Left: critical points of a random plane wave. Center: The Poisson point process which has the same density. Right: a bulk part of the Ginibre ensemble with the same density. For all three point processes depicted in Figure 1 the number of points in a square of side-length |$n$| is |$c\cdot n^2$| where |$c=1/2\sqrt{3}\pi$|⁠. This value of |$c$| is the natural intensity of critical points (see Proposition 1.1) of |$\Psi$|⁠, whereas the other two point processes are so rescaled. The fluctuations of the total number of points in a square depend a lot on the point process. Though formally stated for random spherical harmonics (which are only equivalent to |$\Psi$| in the limit, under a natural scaling), it is likely that one may deduce from [5] that the variance for the critical points scales like |$n^2\log(n)$|⁠, whereas for the Poisson point process it is asymptotic to |$c\cdot n^2$| (with the same |$c$| as above), and for the Ginibre ensemble it is of order |$n$|⁠. On the local scale, the probability that there is at least one point in a small disc or radius |$\rho$| is the same for all three processes due to the translation invariance and our choice of normalization. The respective probabilities that there are exactly two points in a small disc are very different though. For the Poisson point process it is the probability that a Poisson random variable with intensity |$c\pi \rho^2$| is equal to |$2$|⁠. By the definition it is given by \[ \mathbb{P}(2\text{ points})=\frac{\left(c\pi \rho^2\right)^2\exp\left(-c\pi\rho^2\right)}{2}\approx \frac{c^2\pi^2\cdot \rho^4}{2}=\frac{1}{2^3 3}\cdot \rho^4, \] whereas for the Ginibre ensemble (which is a determinantal point process) this probability is of order |$\rho^6$|⁠. That means that the points corresponding to the Ginibre ensemble repel each other, inducing on their visible regularity or rigidity. Our principal result (Theorem 1.2) is the evaluation of the |$2$|nd factorial moment of the number of critical points of |$\Psi$| in a radius |$\rho$| disc, asymptotically for |$\rho\rightarrow 0$|⁠. This suggests (Corollary 1.3 and Conjecture 1.4) that the probability of having precisely two critical points in the disc is $$ \frac{1}{2^6 3\sqrt{3}}\rho^4+o\left(\rho^4\right)\!, $$ of the same order of magnitude (and leading constant smaller by the factor |$8\sqrt{3}\approx 13.8$|⁠) as the probability of finding |$2$|-points in the same disc for the Poisson point process. This minor difference could not stand for the striking difference in the appearance of the two processes, highly regular for the critical points of |$\Psi$|⁠. It is also worth noting, that despite the fact that critical points are more “lattice like” than the Ginibre ensemble, for the critical points clustering is significantly more likely (see Figure 2). Fig. 2. Open in new tabDownload slide Central fragment of the critical points process from Figure 1. Here we distinguish between different types of critical points: diamonds are local maxima, squares are local minima, and discs are saddles. Fig. 2. Open in new tabDownload slide Central fragment of the critical points process from Figure 1. Here we distinguish between different types of critical points: diamonds are local maxima, squares are local minima, and discs are saddles. A possible explanation for this effect could come from the second part of Theorem 1.2. Let us separate the critical point process into two parts, namely extrema and saddles. Both processes are very “regular” and exhibit a strong repulsion. In both cases the probability of having at least two points in a small disc of radius |$\rho$| decays at rate of at least |$\rho^7\log(1/\rho))$| which is smaller than the corresponding decay for the Ginibre ensemble almost by an order of magnitude. We believe that the apparent “rigid” structure that is observed in Figure 1 (left) comes from the regularity of both these point processes. Moreover, it seems that both processes have a very similar structure (seeFigure 3). Fig. 3. Open in new tabDownload slide The same central fragment as in Figure 2 according to their type. Left: extrema only, right: saddles only. Fig. 3. Open in new tabDownload slide The same central fragment as in Figure 2 according to their type. Left: extrema only, right: saddles only. All clustering comes from the probability that after overlapping, a point in one process is close to a point in another process. “Rigidity” and similarity in the structure suggest that the pairs of critical points that are close to each other are well separated and do not affect the general impression of “rigidity” in Figure 1 (left). To formulate our main results we introduce the following notation for the number of critical points of a random plane wave |$\Psi$| in a disc |$\cal{B}(\rho)$| of radius |$\rho>0$|⁠: $$ {\cal N}^c_{\rho}=\# \{x \in \cal{B}(\rho): \nabla \Psi(x)=0\}. $$ The numbers |${\cal N}^{saddle}_{\rho}$|⁠, |${\cal N}^{min}_{\rho}$|⁠, |${\cal N}^{max}_{\rho}$|⁠, and |${\cal N}^{e}_{\rho}$| of saddles, minima, maxima, and extrema respectively may also be defined. Since the function |$\Psi$| is translation invariant, the above random variables are independent of the center of the disc, so for simplicity, we may assume that it is centred at the origin. Another useful observation is that the random plane waves are scale invariant (that is, the law of |$\Psi$| with arbitrary |$k$| on |$\cal{B}(1)$| is (up to homothety) equivalent to the law of |$\Psi$| with |$k=1$| on |$\cal{B}(k)$|⁠); hence, with no loss of generality, we may assume that |$k=1$|⁠, as we will for the rest of this manuscript. The following principal results of this manuscript evaluate the expectation and the second factorial moment of |${\cal N}^c_{\rho}$| for small values of |$\rho$|⁠. The first result, consistent to a similar statement from [4, Proposition 1.1], is for the expectations. Proposition 1.1. For every |$\rho >0$| we have \begin{equation} \label{eq: first moment} \mathbb{E}\left[{\cal N}^c_{\rho}\right]= \frac{1}{2 \sqrt{3}} \rho^2 \end{equation} (3) and \begin{equation*} 4\mathbb{E}\left[{\cal N}^{min}_{\rho}\right]=4\mathbb{E}\left[{\cal N}^{max}_{\rho}\right]=2\mathbb{E}\left[{\cal N}^{saddle}_{\rho}\right]=2\mathbb{E}\left[{\cal N}^{e}_{\rho}\right]=\mathbb{E}\left[{\cal N}^{c}_{\rho}\right]\!. \end{equation*} □ Equation (3) is not an asymptotic result, but rather a precise identity. More generally, same proof works on any open domain |$\Omega$|⁠, that is the expected number of critical points lying in |$\Omega$| is equal to |$\frac{\mathrm{Area}(\Omega)}{2 \sqrt{3}\pi}$|⁠. Evaluating the second moment is more involved, and we were unable to obtain a precise expression. Instead, we show how it behaves asymptotically as the radius |$\rho \to 0$|⁠. Theorem 1.2. As |$\rho \to 0$|⁠, we have the following expansion for the number of critical points: \begin{equation} \label{eq:second moment} \mathbb{E}\left[{\cal N}^c_{\rho} \; \left({\cal N}^c_{\rho}-1\right)\right] = \frac{1}{2^5 \, 3 \, \sqrt 3 } \; \rho^4+O\left(\rho^6\right)\!. \end{equation} (4) For |${\cal N}^{saddle}_{\rho}$|⁠, |${\cal N}^{min}_{\rho}$|⁠, |${\cal N}^{max}_{\rho}$|⁠, and |${\cal N}^{e}_{\rho}$| – numbers of saddles, local minima, local maxima, and local extrema in a ball of radius |$\rho$| we have \begin{align} \label{eq:max-max moment} &\mathbb{E}\left[{\cal N}^{max}_{\rho}({\cal N}^{max}_{\rho}-1)\right]=\mathbb{E}\left[{\cal N}^{min}_{\rho}({\cal N}^{min}_{\rho}-1)\right]=O\left(\rho^7\log(1/\rho)\right)\!, \\ \end{align} (5) \begin{align} \label{eq:extremum-extremum} & \mathbb{E}\left[{\cal N}^{e}_{\rho}\left({\cal N}^{e}_{\rho}-1\right)\right]=O\left(\rho^7\log(1/\rho)\right)\!, \\ \end{align} (6) \begin{align} \label{eq:saddle-saddle moment} &\mathbb{E}[{\cal N}^{saddle}_{\rho}({\cal N}^{saddle}_{\rho}-1)]=O(\rho^7\log(1/\rho)), \\ \end{align} (7) \begin{align} \label{eq:max-min moment} &\mathbb{E}[{\cal N}^{max}_{\rho}{\cal N}^{min}_{\rho}]=O(\rho^{12}), \\ \end{align} (8) \begin{align} \label{eq:extremum-saddle moment} &\mathbb{E}[{\cal N}^{e}_{\rho}{\cal N}^{saddle}_{\rho}]=\frac{1}{2^6 \, 3 \, \sqrt 3 } \; \rho^4+O(\rho^6). \end{align} (9) □ There is no evidence that the estimates (5)–(7), and (8) are sharp. In fact, it seems quite likely, that they are not, for (8) particularly. Since the extremum-saddle covariance (9) gives the main contribution to (4), the last formula (9) is an asymptotic and as such gives a sharp decay rate. For integer-valued random variables it is more natural to consider factorial moments instead of the usual moments. The asymptotic behaviour of the variance, dominated by the expectation (and hence less useful), can be easily obtained by combining (3) and (4) $$ \operatorname{Var}\left[{{\cal N}^c_{\rho}}\right]=\frac{1}{2 \sqrt{3}} \rho^2-\frac{8\sqrt{3}-1}{2^53\sqrt{3}}\rho^4+\dots. $$ Since all our random variables |${\cal N}_\rho$| are integer valued, the first and second factorial moments yield the asymptotics for probabilities of the events |${\cal N}_\rho=1$| and |${\cal N}_\rho\ge 2$|⁠, as follows. Corollary 1.3. As |$\rho \to 0$| we have the following asymptotic formulas for probabilities to have exactly one point: \begin{equation} \label{eq:prob(crit=1)} \begin{aligned} &\mathbb{P}\left({\cal N}^c_{\rho}=1\right)= \frac{1}{2 \sqrt{3}} \rho^2 + O\left(\rho^4\right)\!, \\ &\mathbb{P}\left({\cal N}^{min}_{\rho}=1\right)=\mathbb{P}\left({\cal N}^{max}_{\rho}=1\right)= \frac{1}{8 \sqrt{3}} \rho^2 + O\left(\rho^4\right)\!, \\ &\mathbb{P}\left({\cal N}^{e}_{\rho}=1\right)= \frac{1}{4 \sqrt{3}} \rho^2 + O\left(\rho^4\right)\!, \\ &\mathbb{P}\left({\cal N}^{saddle}_{\rho}=1\right)= \frac{1}{4 \sqrt{3}} \rho^2 + O\left(\rho^4\right)\!. \end{aligned} \end{equation} (10) For probabilities to have at least two points we have \begin{equation} \label{eq:two point probabilities} \begin{aligned} &\mathbb{P}\left({\cal N}^c_{\rho}\ge 2\right) = O\left(\rho^4\right)\!, \\ &\mathbb{P}\left({\cal N}^{min}_{\rho}\ge 2\right) = \mathbb{P}\left({\cal N}^{max}_{\rho}\ge 2\right) =O\left(\rho^7 \log(1/\rho)\right)\!, \\ &\mathbb{P}\left({\cal N}^{saddle}_{\rho}\ge 2\right) =O\left(\rho^7 \log(1/\rho)\right)\!, \\ &\mathbb{P}\left({\cal N}^{min}_{\rho}\ge 1, {\cal N}^{max}_{\rho}\ge 1\right)=O\left(\rho^{12}\right) \end{aligned} \end{equation} (11) Finally, for the probability to have three points we have \begin{equation} \label{eq:three point probability} \mathbb{P}\left({\cal N}^c_{\rho}\ge 3\right)=O\left(\rho^7 \log(1/\rho)\right)\!. \end{equation} (12) □ Proof. The proof is straightforward. For the sake of notational convenience we write |$N={\cal N}^c_{\rho}$|⁠. The expectation and the second factorial moment could be written as series \begin{align*} \mathbb{E}\left[{N}\right]&=\mathbb{P}(N=1)+2\mathbb{P}(N=2)+3\mathbb{P}(N=3)+\dots\\ \mathbb{E}\left[{N(N-1)}\right]&=2\mathbb{P}(N=2)+6\mathbb{P}(N=3)+12\mathbb{P}(N=4)+\dots \end{align*} Comparing the coefficients in front of |$\mathbb{P}(N=k)$| we see that if the second moment is o-small of the expectation, then the expectation is dominated by |$\mathbb{P}(N=1)$|⁠. In this case |$\mathbb{P}(N=1)$| has the same leading term as the expectation and the error term is of the same order as the second moment. This, combined with the results of Proposition 1.1, proves formulas (10). The estimates (11) are obtained by applying Markov inequality to the results of Theorem 1.2. Finally, to prove (12), we notice that the event |${\cal N}^c\ge 3$| is majorized by |${\cal N}^{e}\ge 2$| or |${\cal N}^{saddle}\ge 2$|⁠.■ The third factorial moment should be dominated by the event |$N=3$|⁠, which, by (12), is |$O(\rho^7\log(1/\rho))$|⁠. This gives a strong evidence that $$ \mathbb{E}\left[{N(N-1)(N-2)}\right]=o\left(\rho^4\right)\!. $$ Assuming that, this is indeed true, we can repeat the argument above and compare the coefficients in the second and third factorial moments and show that $$ \mathbb{P}(N=2)=\rho^4/2^6 3\sqrt{3}+o\left(\rho^4\right)\!. $$ 1.3 Outline of the proofs The proofs of Proposition 1.1 and Theorem 1.2 are based on the Kac-Rice formula applied to the gradient of |$\Psi$|⁠. The Kac-Rice formula is a standard tool for studying the expected number of zeros of a random field (see e.g., [1, Theorem 11.2.1] or [2, Theorem 6.8]) and its higher moments by expressing the |$n$|-th (factorial) moment in terms of an |$n$|-dimensional integral. In general, under some non-degeneracy conditions on the given random field, for every |$n\ge 1$| the factorial moments are given by: \begin{align} \label{KRn} \mathbb{E}\left[ {\cal N}^c_{\rho} \left({\cal N}^c_{\rho}-1\right) \cdots \left({\cal N}^c_{\rho}-(n-1)\right)\right]= \idotsint_{{\cal B}(\rho) \times \cdots \times {\cal B}(\rho) } K_n({\bf z}) \; d {\bf z}, \end{align} (13) where |${\bf z}=(z_1, \dots z_n) \in {\cal B}(\rho) \times \cdots \times {\cal B}(\rho) \subset \mathbb{R}^{2n}$|⁠, and |$K_n$| is the |$n$|-point correlation function defined as the conditional Gaussian expectation \begin{equation*} %\label{kkn} K_n({\bf z})=\phi_{(\nabla \Psi(z_1), \dots, \nabla \Psi(z_n))} (0, \dots, 0) \cdot {\mathbb E} \left[ \prod_{i=1}^n |\text{det} H_{\Psi} (z_i)| \; \big| \nabla \Psi(z_1)= \cdots= \nabla \Psi(z_n)=0 \right]\!, \end{equation*} where |$\phi_{(\nabla \Psi(z_1), \dots, \nabla \Psi(z_n))}(0, \dots, 0)$| is the density function of the Gaussian vector $$(\nabla \Psi(z_1), \dots, \nabla \Psi(z_n))$$ evaluated at |$(0, \dots, 0)$|⁠, and |$H_{\Psi} (z_i)$| is the Hessian matrix of |$\Psi$| at |$z_i$|⁠. The Kac-Rice formula in (13) holds under the condition that the Gaussian vector |$(\nabla \Psi(z_1), \dots, \nabla \Psi(z_1))$| is non degenerate. For |$n=1$| the computation of |$K_1$| is straightforward; it is essentially the same as in [4]. We give it below since demonstrates the use of the Kac-Rice formula. The case |$n=2$| is more involved and the asymptotics of |$K_{2}(z_{1},z_{2})$| as |$z_2\to z_1$| (inducing on the second factorial moment) was entirely unexpected. Based on the above computer simulations one would expect for the critical points repel, that is as |$z_2\to z_1$|⁠, |$K_2(z_1,z_2)\to 0$|⁠. That would have indicated that the second factorial moment is |$o(\rho^4)$|⁠, with plausible true order |$\rho^5$| of decay. To our surprise, a precise analysis of the relevant Gaussian integrals have shown that |$K_2$| does not vanish on the diagonal; it has a finite, non-zero limit. Hence the second factorial moment for small |$\rho$| behaves like a constant times the square of the area of |${\cal B}(\rho)$|⁠, that is a constant times |$\rho^{4}$|⁠. It is theoretically possible to compute the behaviour of the higher correlation functions |$K_n$| near the diagonal that is when |$z_i\to z_j$| for |$i\ne j$|⁠, but seems extremely technically demanding. On the other hand, it is easy to believe, that |$K_n$| should stay bounded. Considering it as a given and using the same argument as in the proof of Corollary 1.3, we obtain the following conjecture. Conjecture 1.4. For |$n>2$| and |$\rho\to 0$| we have the following estimate of the factorial moment $$ \mathbb{E}\left[ {\cal N}^c_{\rho} \left({\cal N}^c_{\rho}-1\right) \cdots \left({\cal N}^c_{\rho}-(n-1)\right)\right]=O\left(\rho^{2n}\right) $$ and, correspondingly, on the probability to have exactly |$n$| points in a small ball \[ \mathbb{P}\left({\cal N}^c_{\rho}=n\right)=O\left(\rho^{2n}\right)\!. \] □ Be believe that this estimate holds, but we know that it is not sharp since already for |$n=3$| we have |$\mathbb{P}({\cal N}^c_{\rho}=3)=O(\rho^{7}\log(1/\rho))=o(\rho^6)$| (see (12)). 2 Expected Number of Critical Points 2.1 On the Kac-Rice formula for computing the expected number of critical points In this section, we apply Kac-Rice formula to compute the expected value of |${\cal N}_{\rho}^c$|⁠. Counting the critical points of |$\Psi$| in the ball |${\cal B}(\rho)$| is equivalent to counting the zeros of the map |${\cal B}( \rho) \to {\mathbb R}^2$| given by |$z \to \nabla \Psi(z)$|⁠. One defines the zero density |$K_1: {\cal B}(\rho) \to \mathbb{R}$| of |$\Psi$| as $$K_1(z)=\phi_{\nabla \Psi(z)} (0,0) \cdot \mathbb{E}[|\mathrm{det}\, H_{\Psi}(z)| \big| \nabla \Psi(z)=0],$$ where |$\phi_{\nabla \Psi(z)}$| is the Gaussian probability density of two-dimensional vector |$\nabla \Psi(x) \in \mathbb{R}^2$| evaluated at |$(0,0)$|⁠, and |$H_{\Psi}(z)$| is the Hessian matrix of |$\Psi$| at |$z$|⁠. By the Kac-Rice formula, if |$\nabla \Psi(z)$| is nonsingular for all |$z \in {\cal B}(\rho)$|⁠, then \begin{align} \label{KRe} \mathbb{E}[{\cal N}^c_{\rho}]=\int_{\cal{B}(\rho)} K_1(z) d z. \end{align} (14) 2.2 Proof of Proposition 1.1 We first observe that in our case the zero density |$K_1$| is independent of |$z$| because |$\Psi$| is isotropic; hence the Kac-Rice formula (14) states that, \begin{align} \label{cont3} \mathbb{E}[{\cal N}^c_{\rho}]=\pi \rho^2 K_1. \end{align} (15) Moreover, as we are dealing with a smooth Gaussian field, it is possible to write an analytic expressions for |$K_1$| in terms of the covariance function |$\psi$| and its derivatives; to derive such analytic expression we evaluate the covariance matrix |$\Sigma$| of the |$5$|-dimensional centred jointly Gaussian vector $$(\nabla \Psi(z), \nabla^2 \Psi(z))$$ where |$ \nabla^2 \Psi(z)$| is the vectorized Hessian evaluated at |$z$|⁠, that is a vector $$(\partial^2_{z_1,z_1}\Psi(z),\partial^2_{z_1,z_2}\Psi(z),\partial^2_{z_2,z_2}\Psi(z)).$$ The covariance matrix |$\Sigma$| of |$(\nabla \Psi(z), \nabla^2 \Psi(z))$| is evaluated in Appendix B.1 and has the form \[ \Sigma=\left(\begin{array}{ccc} A & B\\ B^t& C \end{array} \right)\!, \] where, \begin{align} \label{matrices} A= \left(\begin{array}{cc} \frac{1}{2} &0 \\ 0&\frac{1}{2} \end{array} \right), B=0, C= \left( \begin{array}{ccc} \frac{3 }{8} &0& \frac{1}{8} \\ 0&\frac{1}{8}&0\\ \frac{1}{8}&0&\frac{3}{8} \end{array}\right)\!. \end{align} (16) From |$A$| we immediately obtain the probability density of the |$two$|-dimensional vector |$\nabla \Psi(z)$| evaluated at |$(0,0)$|⁠: \begin{align} \label{cont2} \phi_{\nabla \Psi(z)} (0,0)=\frac{1}{2 \pi \sqrt{ 1/4} }=\frac{1}{\pi}, \end{align} (17) in addition, since the first and the second order derivatives of |$\Psi$| are independent at every fixed point |$z \in \mathbb{R}^2$|⁠, we have that \[ \mathbb{E}[|\mathrm{det} H_{\Psi}(z)| \big| \nabla \Psi(z)=0]=\mathbb{E}[|\mathrm{det} H_{\Psi}(z)|]. \] From the covariance matrix |$C$| of |$\nabla^2 \Psi(z)$| in (16) we immediately see that \begin{align} \label{ev} \mathbb{E}[|\mathrm{det} H_{\Psi}(z)|]= \frac{1}{8} \mathbb{E}\left[|Y_1 Y_3 -Y_2^2|\right]\!, \end{align} (18) where |$Y=(Y_{1},Y_{2},Y_{3})$| is a centred jointly Gaussian random vector with covariance matrix \begin{equation*} C_1=\left( \begin{array}{ccc} 3 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 3 \end{array} \right)\!. \end{equation*} To evaluate (18) we introduce the transformation |$W_1=Y_1$|⁠, |$W_2=Y_2$|⁠, |$W_3=Y_1+Y_3$|⁠, and we write |$\mathbb{E}[|Y_1 Y_3 -Y_2^2|]$| in terms of a conditional expectation as follows \begin{align} \label{ccc} \mathbb{E}\left[|Y_1 Y_3 -Y_2^2|\right]= \mathbb{E}_{W_3} \left[ \; \mathbb{E}\left[|W_1 W_3-W_1^2 - W_2^2 | \big| W_3=t \right] \; \right]; \end{align} (19) to evaluate the conditional expectation in (19) we follow the argument in the proof of [4, Proposition 1.1], that is we note that \begin{align*} \mathbb{E}[|W_1 W_3-W_1^2 - W_2^2 | \big| W_3=t]& =\mathbb{E}\left[ \left. \left\vert W_{1} \, t -W_{1}^{2}-W_{2}^{2}% \right\vert \right\vert W_{3}= t \right] \\ &=\mathbb{E}\left[ \left\vert (Z_{1}+ {t}/{2})\, t -(Z_{1}+ {t}/{2})^{2}-Z_{2}^{2}\right\vert \right] \\ &=\mathbb{E}\left[ \left\vert -Z_{1}^{2}-Z_{2}^{2}+t^{2}/4\right\vert \right]=\mathbb{E} \left[ \left| - X+\frac{t^2}{4} \right| \right]\!, \end{align*} where |$Z_1, Z_2$| are independent standard Gaussian and |$X$| is a |$\chi$|-squared random variable with density \begin{align*} f_X(x)=\frac{1}{2}e^{-\frac{x}{2}}, \ x >0. \end{align*} It follows that \begin{align*} \mathbb{E} \left[ \left| \frac{t^2}{4} - X \right| \right] = -2+4 e^{-\frac{t^2}{8}}+\frac{t^2}{4}, \end{align*} and \begin{align} \label{cont1} \mathbb{E}[|Y_1 Y_3 -Y_2^2|]=\frac{1}{4 \sqrt \pi} \int_{\mathbb{R}} e^{-\frac{t^2}{16}} \left( -2+4 e^{-\frac{t^2}{8}}+\frac{t^2}{4} \right) d t = \frac{2^2}{\sqrt 3 }. \end{align} (20) The statement follows combining (15), (17), (20), and observing that \begin{align*} \mathbb{E}[{\cal N}^c_{\rho}(\Psi)]=\pi \rho^2 \cdot \frac{1}{\pi} \cdot \frac{1}{8} \frac{2^2}{\sqrt 3 }= \frac{1}{2 \sqrt 3 }\cdot \rho^2. \end{align*} 3 Second Factorial Moment 3.1 On the Kac-Rice formula for computing the second factorial moment of the number of critical points We will find an explicit expression for the |$2$|-point correlation function |$K_2 : {\cal B}(\rho) \times {\cal B}(\rho) \to \mathbb{R}$|⁠, defined as the conditional Gaussian expectation \begin{align*} K_2(z,w)=\phi_{(\nabla \Psi(z),\nabla \Psi(w))} (0, 0) \cdot \mathbb{E}\left[ |\mathrm{det} H_{\Psi}(z)| \cdot |\mathrm{det} H_{\Psi}(w)| \big| \nabla \Psi(z)=\nabla \Psi(w)=0\right]\!, \end{align*} in terms of the covariance function |$\psi$| and its derivatives. Finding such an expression involves studying the centred Gaussian vector \begin{equation} \label{repp} \left(\nabla \Psi(z),\nabla \Psi(w),\nabla^2 \Psi(z),\nabla^2 \Psi(w)\right) \end{equation} (21) with covariance matrix |${\bf \Sigma}(z,w)$|⁠, |$z,w \in {\cal B}(\rho)$|⁠. It is known [2, Theorem 6.9] that, if for all |$z \ne w$| the Gaussian distribution of |$(\nabla \Psi(z),\nabla \Psi(w))$| is non-degenerate, the second factorial moment of the number of critical points in |${\cal B}(\rho)$| can be expressed as \begin{align} \label{18:03} \mathbb{E}\left[{\cal N}^c_{\rho} \; \left({\cal N}^c_{\rho}-1\right)\right]=\iint_{\cal{B}(\rho) \times \cal{B}(\rho)} K_2(z,w) \; d z \, d w. \end{align} (22) We note that |$K_2$| is everywhere nonnegative. 3.2 Proof of Theorem 1.2 In order to study the asymptotic behaviour of the second factorial moment of the number of critical points in |${\cal B}(\rho)$|⁠, as the radius |$\rho$| of the disk goes to zero, we need to study the centred Gaussian random vector (21). Its covariance matrix |${\bf \Sigma}={\bf \Sigma}(z,w)$| is of the form $${\bf \Sigma}=\left(\begin{array}{ccc} {\bf A} & {\bf B}\\ {\bf B}^t& {\bf C} \end{array} \right)\!, $$ where |${\bf A}={\bf A}(z,w)$| is the covariance matrix of the gradients |$(\nabla \Psi(z),\nabla \Psi(w))$|⁠, |${\bf C}={\bf C}(z,w)$| is the covariance matrix of the second order derivatives |$(\nabla^2 \Psi(z),\nabla^2 \Psi(w))$| and |${\bf B}={\bf B}(z,w)$| is the covariance matrix of the first and second order derivatives. The function |$\Psi$| is isotropic, hence, the critical point process is also invariant w.r.t. translations and rotations. This means that its |$2$|-point function |$K_2(z,w)$| depends on |$|z-w|$| only (this is not true for covariance matrix |$\bf \Sigma$|⁠); by the standard abuse of notation we write \begin{equation} \label{eq:K2=K2(d)} K_2(z,w)=K_2(|z-w|). \end{equation} (23) We will compute |$K_2(z,w)$| for |$z=(0,0)$| and |$w=(0,r)$|⁠, which, thanks to the by-product (23) of the isotropic property of |$\Psi$|⁠, this will give us |$K_2(r)$|⁠. In Appendix B.2 we evaluate the entries of |${\bf \Sigma}(z,w)$| with |$z$| and |$w$| as above, and in Appendix B.3 we evaluate the covariance matrix |${\bf \Delta}={\Delta}(z,w)$| of |$(\nabla^2 \Psi(z),\nabla^2 \Psi(w))$| conditioned on |$\nabla \Psi(z)=\nabla \Psi(w)=0$|⁠, that is \begin{align*} {\Delta}={\bf C} - {\bf B}^t {\bf A}^{-1} {\bf B}. \end{align*} From now on we will work only with |${\bf \Sigma}(r)$| and |${\Delta}(r)$| which we define as |${\bf \Sigma}(z,w)$| and |${\Delta}(z,w)$| with |$z=(0,0)$| and |$w=(0,r)$|⁠. As we discussed above, the two-point function is given by \begin{equation} \label{k2} \begin{aligned} K_2(r)&=\frac{1}{(2 \pi)^2 \sqrt{\text{det}({\bf A}(r))}} \\ &\quad\times \int_{\mathbb{R}^6 }\left|\zeta_{1} \zeta_{3} - \zeta^2_{2}\right| \cdot \left|\zeta_{4} \zeta_{6} - \zeta^2_{5}\right| \frac{1}{(2 \pi)^3} \frac{1}{\sqrt{\text{det}({\Delta}(r))}} \exp \left\{-\frac{1}{2} \zeta^t {\Delta}^{-1}(r) \zeta \right\} d \zeta, \end{aligned} \end{equation} (24) where |$\zeta=(\zeta_1, \zeta_2, \zeta_3, \zeta_4, \zeta_5, \zeta_6)$| is a vector in |$\mathbb{R}^6$|⁠. Indeed, the density of |$(\nabla \Psi(0,0),\nabla\Psi(0,r))$| at zero is given by |$(2\pi)^{-2}(\det({\bf A}(r)))^{-1/2}$|⁠, and the integral gives the expectation of |$|\mathrm{det} H_{\Psi}(z)| \cdot |\mathrm{det} H_{\Psi}(w)|$| with respect to the Gaussian measure of |$(\nabla^2 \Psi(z),\nabla^2\Psi^2(w))$| conditioned on |$\nabla \Psi(z)=\nabla \Psi(w)=0$|⁠, that is, having covariance |${\Delta}(r)$|⁠. Our aim is to study the asymptotic behaviour of the |$2$|-point correlation function |$K_2$| in the vicinity of |$r=0$|⁠. For every strictly positive |$r$|⁠, |${\Delta}(r)$| is symmetric, hence we may diagonalize it with an orthogonal |$P(r)$|⁠: \begin{align} \label{18:07} {\Delta}(r)=P^{-1}(r) \Lambda(r) P(r)=P^t(r) \Lambda(r) P(r), \end{align} (25) where the matrix |$\Lambda(r)$| is diagonal, with eigenvalues |$\lambda_i(r)$|⁠, |$i=1, \dots,6$|⁠, and |$P(r)$| is the orthogonal matrix with row vectors the normalized eigenvectors of |${\Delta}(r)$|⁠. The analytic expressions of the eigenvalues and eigenvectors of |${\Delta}(r)$|⁠, |$r>0$|⁠, are computed in Lemma A.1 and Lemma A.3 respectively. In Lemma A.2 and Lemma A.4 we compute their Taylor expansion around |$r=0$|⁠. We prove these lemmas with the aid of Mathematica since the calculations are technically demanding. We stress that all the computations performed with Mathematica are symbolic. Equation (25) implies that we can write \begin{equation} \label{13:05} \begin{aligned} &\frac{1}{\sqrt{\text{det}({\Delta}(r))}}\exp \Big\{ -\frac{1}{2} \zeta^t {\Delta}^{-1}(r) \zeta \Big\}& \\ &= \frac{1}{\sqrt{\prod_{i=1}^6 \lambda_i(r)}}\exp \Big\{ -\frac{1}{2} \zeta^t P^{-1}(r) \Lambda^{-1}(r) P(r) \zeta \Big\} \\ &=\frac{1}{\sqrt{\prod_{i=1}^6 \lambda_i(r)}}\exp\Big\{ -\frac{1}{2} (\Lambda^{-1/2}(r)P(r) \zeta)^t (\Lambda^{-1/2}(r)P(r) \zeta) \Big\}. \end{aligned} \end{equation} (26) This suggests to introduce a new variable |$\xi=\Lambda^{-1/2}(r) P(r) \zeta$|⁠. Clearly, we can express |$\zeta$| in terms of |$\xi$| as \begin{equation} \label{eq: zeta} \zeta=P^{-1}(r)\Lambda^{1/2}(r)\xi=P^{t}(r)\Lambda^{1/2}(r)\xi \end{equation} (27) With this change of variables $$ \frac{1}{\sqrt{\text{det}({\Delta}(r))}}\exp \Big\{ -\frac{1}{2} \zeta^t {\Delta}^{-1}(r) \zeta \Big\}d \zeta= e^{-|\xi|^2/2}d\xi. $$ Using (27), we can write components |$\zeta_i$| as $$ \zeta_i=\sum_{j=1}^6 (Q(r))_{ij}\; \sqrt{\lambda_j(r)} \; \xi_j=\sum_{j=1}^6 q_{ij}(r)\; \sqrt{\lambda_j(r)} \; \xi_j, $$ where the |$q_{i j}(r)$| are the elements of |$Q(r)=P^{-1}(r)=P^{t}(r)$|⁠. The columns of |$Q$| form an orthonormal basis of the eigenvectors of |${\Delta}(r)$|⁠. With this change of variables we can rewrite the two quadratic forms |$\zeta_{1} \zeta_{3}- \zeta_{2}^2$| and |$\zeta_{4} \zeta_{6}- \zeta_{5}^2$| in (24) as \begin{align*} & {{\zeta }_{1}}{{\zeta }_{3}}-\zeta _{2}^{2}=\left( \sum\limits_{j=1}^{6}{{{q}_{1j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)\left( \sum\limits_{j=1}^{6}{{{q}_{3j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)-{{\left( \sum\limits_{j=1}^{6}{{{q}_{2j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)}^{2}}, \\ & {{\zeta }_{4}}{{\zeta }_{6}}-\zeta _{5}^{2}=\left( \sum\limits_{j=1}^{6}{{{q}_{4j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)\left( \sum\limits_{j=1}^{6}{{{q}_{6j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)-{{\left( \sum\limits_{j=1}^{6}{{{q}_{5j}}}(r)\sqrt{{{\lambda }_{j}}(r)}\ {{\xi }_{j}} \right)}^{2}}. \\ \end{align*} Summing it all up, the |$2$|-point correlation function |$K_2$| in (24) in |$\xi$| coordinates becomes \begin{equation} \label{k2ch} \begin{aligned} &K_2(r)= \frac{1}{ (2\pi)^5 \sqrt{ \text{det}({\bf A}(r)) }} \int\limits_{\mathbb{R}^6}|\zeta_{1} \zeta_{3} - \zeta^2_{2}| \cdot |\zeta_{4} \zeta_{6} - \zeta^2_{5}| \, \exp \Big\{-\frac{1}{2} \sum_{i=1}^6 \xi^2_i \Big\} d \xi \end{aligned} \end{equation} (28) where |$\zeta_{1} \zeta_{3} - \zeta^2_{2}$| and |$\zeta_{4} \zeta_{6} - \zeta^2_{5}$| are functions of |$\xi_i$| as described above. To obtain the asymptotic behaviour around|$r=0$| of the integral in (28), we Taylor expand around the origin the entries |$q_{i j}$| of the matrix |$Q$| and eigenvalues |$\lambda_j$|⁠. Such Taylor expansions up to |$O(r^4)$| are given by equations (A3) and (A4). Combining these expansions and noting that the first two factors in the integrand are homogeneous polynomials of degree |$2$| in terms of |$\xi$| we obtain the following expansion: \begin{align*} &\Big[ \sum_{j} q_{1 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j \sum_{j} q_{3 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j - \Big(\sum_{j} q_{2 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j \Big)^2 \Big] \\ &\qquad\times \Big[ \sum_{j} q_{4 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j \sum_{j} q_{6 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j - \Big(\sum_{j} q_{5 j}(r) \sqrt{\lambda_{j}(r) } \; \xi_j \Big)^2 \Big]\\ &\quad= - \frac{1}{2^7 3} \xi_4^2 \xi_6^2 r^2 + (1+ ||\xi||^4)\; O(r^4 ), \end{align*} and then \begin{align} \label{14:00} K_2(r)%&= \frac{1}{ (2\pi)^5 \sqrt{ \text{det} A(r) }} \iint_{\mathbb{R}^3 \times \mathbb{R}^3} %\big| h(r,\xi) \big| \times \exp \left\{-\frac{1}{2} \sum_{i=1}^6 \xi^2_i \right\} d \xi_1 \cdots d \xi_6\\ &= \frac{1}{ (2\pi)^5 \sqrt{ \text{det}({\bf A}(r)) }} \left[ \frac{r^2}{2^7 3} \int_{\mathbb{R}^6} \xi_4^2 \xi_6^2 \times \exp \left\{-\frac{1}{2} \sum_{i=1}^6 \xi^2_i \right\} d \xi+ O(r^4) \right]\!. \end{align} (29) In the Gaussian integral variables separate and it is a product of standard one-dimensional integrals. Each of them is equal to |$\sqrt{2\pi}$| and the entire integral is |$(2\pi)^3$|⁠. Matrix |$\bf A$| has a simple block structure and it is easy to compute its determinant. Explicit computation in Appendix B.2 (see equation (B4)) gives $$ \det(A)=\frac{3r^4}{2^8}+O(r^6). $$ Combining this asymptotic with (29), we finally obtain that, as |$r \to 0$|⁠, \begin{align*} K_2(r)=\frac{1}{2^5 3 \sqrt 3 \pi^2}+O(r^2), \end{align*} and, in view of (22), as |$\rho \to 0$|⁠, \begin{align*} \mathbb{E}[{\cal N}^c_{\rho} \; ({\cal N}^c_{\rho}-1)]%= \pi \rho^2 2 \pi \int_{0}^{\rho}K_2(r) r d r = \frac{1}{2^5 3 \sqrt 3 \pi^2} \pi^2 \rho^4+O(\rho^6)=\frac{1}{2^5 3 \sqrt 3 } \rho^4+O(\rho^6). \end{align*} To prove the second part of Theorem 1.2 we need to evaluate the two-point correlation function |$K_2$| modified for the respective types of critical points. The modified function |$K_{2}$| has the same expression (24) with the integration over a proper subset of |$\mathbb{R}^6$|⁠, that is the |$\zeta$| with the corresponding critical points of the prescribed types. To be more precise, let us define two Hessians at points |$z$| and |$w$| (already conditioned to be critical points). In terms of |$\zeta_i$| these Hessians are given by \begin{equation*} H_1= \begin{pmatrix} \zeta_1 & \zeta_2 \\ \zeta_2 & \zeta_3 \end{pmatrix}, \quad \mathrm{and} \quad H_2= \begin{pmatrix} \zeta_4 & \zeta_5 \\ \zeta_5 & \zeta_6 \end{pmatrix} \end{equation*} The characteristic polynomials for these matrices are $$ x^2+b_1x+c_1=x^2-(\zeta_1+\zeta_3)x+\zeta_1\zeta_3-\zeta_2^2 $$ and $$ x^2+b_2x+c_2=x^2-(\zeta_4+\zeta_6)x+\zeta_4\zeta_6-\zeta_5^2. $$ The particular type of a critical point depends on the eigenvalues of its Hessian: they are both negative for the local maxima, positive for the local minima, and of different signs for the saddles. We may reformulate these dependencies in terms of the coefficients |$b_i=-\mathrm{ Tr}\, H_i$| and |$c_i=\det(H_i)$|⁠: a critical point with Hessian |$H_i$| is a minimum if |$c_i>0$| and |$b_i<0$|⁠, a maximum if |$c_i>0$| and |$b_i>0$|⁠, and a saddle if |$c_i<0$| (we may ignore the special probability |$0$| cases when one of the eigenvalues vanishes). As before, we rewrite |$\zeta_i$| in terms of |$\xi_i$|⁠. This gives the coefficients of the polynomials as functions of |$\xi_i$| and |$r$|⁠. Expanding in powers of |$r$| we get \begin{align} b_1&=-\frac{\xi_6}{\sqrt{3}}+\left({\frac{\xi_6}{144\sqrt{3}} -\frac{\xi_5}{96\sqrt{2}}}\right)r^2+O\left(r^3\right)=b_{1,0}+b_{1,2}r^2+O\left(r^3\right) \notag\\ c_1&=-\frac{\xi_4\xi_6}{8\sqrt{6}}r+ \frac{-9\xi_1^2-9\xi_4^2+2\sqrt{6}\xi_6\xi_5+4\xi_6^2}{2^7 3^2}r^2+O\left(r^3\right) =c_{1,1}r+c_{1,2}r^2+O\left(r^3\right) \notag\\ b_2&=-\frac{\xi_6}{\sqrt{3}}+\left({\frac{\xi_6}{144\sqrt{3}} -\frac{\xi_5}{96\sqrt{2}}}\right)r^2+O(r^3)=b_{2,0}+b_{2,2}r^2+O\left(r^3\right)\notag\\ c_2&=\frac{\xi_4\xi_6}{8\sqrt{6}}r+ \frac{-9\xi_1^2-9\xi_4^2+2\sqrt{6}\xi_6\xi_5+4\xi_6^2}{2^7 3^2}r^2+O\left(r^3\right) =c_{2,1}r+c_{2,2}r^2+O\left(r^3\right).\label{eq:b1,c1,b2,c2 as on r} \end{align} (30) We observe the following: all of the coefficients |$b_{i,j}$| are linear functions of |$\xi$|⁠, and all of the coefficients |$c_{i,j}$| are quadratic forms. We also notice that $$ b_{1,0}=b_{2,0}, \ \ b_{1,2}=b_{2,2}, \ \ c_{1,1}=-c_{2,1}, \ \ c_{1,2}=c_{2,2}. $$ Since all the expressions we deal with are homogeneous functions of various degrees, it is natural to work in spherical coordinates. We introduce |$s_i= \xi_i/|\xi|$| and rescale |$b_i$| by |$|\xi|$| and |$c_i$| by |$|\xi|^2$|⁠. Abusing notation we denote the rescaled coefficients |$b_i$|⁠, |$b_{i,j}$|⁠, |$c_i$|⁠, and |$c_{i,j}$| that are now functions of |$s_i$| instead of |$\xi_i$| by the same letters; there is no confusion since from now on all expressions will be in terms of |$|\xi|\in (0,\infty)$| and |$s=(s_1,\dots,s_6)\in S^5$|⁠. With this notation, the formula (28) for |$K_2$| becomes \begin{equation} \label{eq:spherical coordinates} \begin{aligned} K_{2}(r)&= \frac{1}{ (2\pi)^5 \sqrt{ \text{det}({\bf A}(r)) }} \int_{\mathbb{R}^6} |\xi|^4 |c_1 c_2| e^{-|\xi|^2/2}d\xi \\ & = \frac{1}{ (2\pi)^5 \sqrt{ \text{det}({\bf A}(r)) }} \int_0^\infty |\xi|^9 e^{-|\xi|^2/2} d |\xi| \int_{S^5}|c_1(s) c_2(s)| ds \\&= \frac{12}{ \pi^5 \sqrt{ \text{det}({\bf A}(r)) }} \int_{S^5}|c_1(s) c_2(s)| ds, \end{aligned} \end{equation} (31) where |$ds$| is the spherical volume element on the unit sphere |$S^5$|⁠, and we evaluated the standard Gaussian integral $$\int_0^\infty |\xi|^9 e^{-|\xi|^2/2} d |\xi| = 2^{7}\cdot 3.$$ Minimum–minimum two point function. The two-point correlation function |$K_2^{min,min}(r)$| corresponding to the local minima is given by (31) except that we replace the domain of the integration |$S^5$|⁠, by $$ S_{min,min}=\{s\in S^5: c_1>0, c_2 >0, b_1<0, b_2<0\}, $$ the set of |$s$| such that both Hessians correspond to local minima. If |$|s_4 s_6|>\mathrm{C} r$| for sufficiently large |$\mathrm{C}$|⁠, then |$c_1$| and |$c_2$| are of opposite signs (for the rest of this section we use |$\mathrm{C}$| to denote all absolute constants). This implies that |$S_{min,min}$| is a subset of |$\{s:|s_4 s_6|<\mathrm{C} r\}$| for some |$\mathrm{C}$| sufficiently big. It is easy to see that on this set |$|c_i|=O(r^2)$|⁠, thus $$ \int_{S_{min,min}}|c_1(s) c_2(s)| ds \le \int_{\{s:|s_4 s_6|<\mathrm{C} r\}}|c_1(s) c_2(s)| ds \le O(r^4)\int_{\{s:|s_4 s_6|<\mathrm{C} r\}}ds = O(r^5\log(1/r)). $$ That yields |$K_2^{min,min}(r) =O(r^3\log(1/r))$| via (31), where |$r^2$| cancelled out with |$\sqrt{\det({\bf A})}$|⁠. Integrating this estimate over |$\cal{B}(\rho)\times \cal{B}(\rho)$| we obtain an estimate of the second factorial moment: $$ \mathbb{E}\left[{{\cal N}_\rho^{min}({\cal N}_\rho^{min}-1)}\right]=O(\rho^7\log(1/\rho)). $$ The other estimate of (5) follows from symmetry considerations. Minimum–maximum two point function. In the similar way, we have to estimate the integral over |$S_{min,max}$|⁠, the set where one point is a minimum and the other is a maximum. This set is given by conditions that both |$c_i$| are positive and |$b_1$| and |$b_2$| are of different signs. First, the same argument as above forces |$|s_4 s_6|<\mathrm{C} r$| for some large constant |$\mathrm{C}$|⁠. If |$|s_6|>\mathrm{C} r^2$| for a large constant |$\mathrm{C}$|⁠, then the leading terms in the formulas (30) for |$b_i$| dominate and |$b_1$| and |$b_2$| are of the same sign, contradicting our assumption. Hence this implies that |$|s_6|<\mathrm{C} r^2$|⁠, and under this assumption both |$b_i$| are of the form $$ -\frac{s_6}{\sqrt{3}}-\frac{s_5 r^2}{96\sqrt{2}}+O(r^3). $$ Again, since |$b_i$| should be of different signs, it forces the term corresponding to |$O(r^3)$| to dominate, that is $$ L(s_5,s_5)=\left|-\frac{s_6}{\sqrt{3}}-\frac{s_5 r^2}{96\sqrt{2}}\right|\le \mathrm{C} r^3, $$ for some big constant |$\mathrm{C}$|⁠. Notice that this condition is stronger than the previous condition that |$|s_6|<\mathrm{C} r^2$|⁠. Substituting the estimate |$|s_6|<\mathrm{C} r^2$| into the formulas (30) for |$c_1$| and |$c_2$| we see that they both are equal to $$ -\frac{9}{2^7 3^2}(s_1^2+s_4^2)r^2+O(r^3). $$ It then follows that |$(s_1^2+s_4^2)$| is bounded by |$\mathrm{C} r$| (for large |$\mathrm{C}$|⁠), as otherwise both |$c_1$| and |$c_2$| are negative; also under this condition both |$c_i$| are |$O(r^3)$|⁠. Combining all of this we get the estimate $$ \int\limits_{S_{min,max}}|c_1(s) c_2(s)| ds \le \int\limits_{\substack{(s_1^2+s_4^2)<\mathrm{C} r \\ L(s_5,s_5)<\mathrm{C} r^3 }}|c_1(s) c_2(s)| ds = O(r^6)\int\limits_{\substack{(s_1^2+s_4^2)<\mathrm{C} r \\ L(s_5,s_5)<\mathrm{C} r^3 }}ds=O(r^6)O(r^4)=O\left(r^{10}\right)\!, $$ and substituting this into (31) and integrating |$K_{2}$| the Kac-Rice formula yields $$ \mathbb{E}\left[{\cal{N}_\rho^{min}\cal{N}_\rho^{max}}\right]=O\left(\rho^{12}\right)\!. $$ Saddle–saddle and extremum–extremum two point functions. For two extrema or two saddle points both |$c_i$| are forced to be of the same sign. The same argument as for the minimum–minimum case yields $$ \mathbb{E}\left[{\cal{N}_\rho^{saddle}(\cal{N}_\rho^{saddle}-1)}\right]= O(\rho^7\log(1/\rho)), $$ and $$ \mathbb{E}\left[{\cal{N}_\rho^{e}(\cal{N}_\rho^{e}-1)}\right]= O(\rho^7\log(1/\rho)). $$ Extremum–saddle two point function. Finally we notice that |$\cal{N}=\cal{N}^{e}+\cal{N}^{saddle}$|⁠, and $$ \cal{N}(\cal{N}-1)=\cal{N}^{e}(\cal{N}^{e}-1)+\cal{N}^{saddle}(\cal{N}^{saddle}-1)+2\cal{N}^{e}\cal{N}^{saddle}. $$ Combining this formula with previous estimates we obtain $$ \mathbb{E}\left[{\cal{N}_\rho^{e}\cal{N}_\rho^{saddle}}\right]=\frac{1}{2}\mathbb{E}\left[{\cal{N}_\rho(\cal{N}_\rho-1)}\right]+O(\rho^7\log(1/\rho)) = \frac{1}{2^6 \, 3 \, \sqrt 3 } \; \rho^4+O\left(\rho^6\right)\!. $$ This completes the proof of Theorem 1.2. Funding The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme [FP7/2007-2013] / ERC grant [agreements no 335141 to I.W. and V.C.] and Engineering and Physical Sciences Research Council (EPSRC) Fellowship [EP/M002896/1 to D.B.]. Acknowledgements We are grateful to Peter Sarnak for some stimulating discussions especially with regards to the proof of Theorem 1.2, and to Robert Adler, Anne Estrade, and Mikhail Sodin for useful conversations. We thank the referees for careful reviews and useful comments. Appendix A: Eigenvalue and eigenvectors of |${\Delta}(r)$|⁠, |$r>0$| We introduce the notation $$ {\Delta} (r)=\left( \begin{array}{cc} \Delta_1(r) & \Delta_2(r) \\ \Delta_2(r) & \Delta_1(r) \end{array} \right) $$ where |$\Delta_1$| and |$\Delta_2$| are |$3\times 3$| symmetric matrices, and define |$a_i$|⁠, |$i=1,\dots 8$| so that \begin{equation} \label{eq:definition delta_i} \begin{aligned} \Delta_1(r) = \left( \begin{array}{ccc} \frac{1}{3} +a_1(r)&0& a_4(r)\\ 0&a_2(r)&0\\ a_4(r)&0&a_3(r) \end{array}\right), \ \Delta_2(r) = \left( \begin{array}{ccc} \frac 1 3 + a_5(r)&0& a_8(r)\\ 0&a_6(r)&0\\ a_8(r)&0&a_7(r) \end{array}\right)\!. \end{aligned} \end{equation} (A1) We compute now the eigenvalues and eigenvectors of the matrix |${\Delta}(r)$|⁠, |$r>0$|⁠. We introduce the following notation: \begin{gather*} A_1^{+}(r)= a_1(r) + a_5(r)+\frac{2}{3}, \ A_1^{-}(r)= a_1(r) - a_5(r),\\ A_2^{\pm}(r)= a_2(r) \pm a_6(r), \\ A_3^{\pm}(r)= a_3(r) \pm a_7(r), \\ A_4^{\pm}(r)= a_4(r) \pm a_8(r); \end{gather*} with |$a_i(r)$| defined above. This lemma expresses the eigenvalues of |$\Delta$| in terms of |$A_i^\pm$| which, in their term, are expressed in terms of |$a_i$|⁠. In (B5) we will compute the asymptotic behaviour of |$a_i$|⁠. Substituting these expansions into explicit formulas (using Mathematica)(A2) we get expansions for |$\lambda_i$| and |$\sqrt{\lambda_i}$|⁠. After obtaining the explicit formulas for eigenvalues we, again, use computer algebra to find explicit formulas for eigenvectors of |$\Delta$|⁠. The elements of |$v_i$| are explicit algebraic expressions in terms of |$a_i$| that are defined by (A1). Normalizing the vectors and using expansions of |$a_i$| (B5) we obtain the following expansion for the matrix |$Q$| Appendix B: Expansions of covariance matrices B.1: Covariance matrix of |$\boldsymbol{(\nabla \Psi(z), \nabla^2 \Psi(z))}$| In this section we compute the covariance matrix |$\Sigma$| of the |$5$|-dimensional centred Gaussian vector which combines the gradient and the elements of the Hessian evaluated at |$z$|⁠. By the translation invariance of |$\Psi$|⁠, |$\Sigma$| does not depend on the point |$z \in \mathbb{R}^2$|⁠. It is convenient to write the covariance matrix in blocks $$\Sigma=\left(\begin{array}{ccc} A & B\\ B^t& C \end{array} \right)\!,$$ where \begin{align*} A=\mathbb{E}[ \nabla \Psi(z)^t \cdot \nabla \Psi(z)], \ B=\mathbb{E}[ \nabla \Psi(z)^t \cdot \nabla^2 \Psi(z)], \ C=\mathbb{E}[ \nabla^2 \Psi(z)^t \cdot \nabla^2 \Psi(z)]. \end{align*} It is a standard fact that covariances of the derivative are given by derivatives of the covariance kernel. The computations of |$A$|⁠, |$ B$| and |$C$| are quite lengthy, but they do not require sophisticated arguments other than iterative differentiation of Bessel functions. For example, to compute |$A$| we first have to compute expressions \begin{equation*} \mathbb{E}[ \partial_{z_i} \Psi(z) \; \partial_{w_j} \Psi(w) ] = \frac{\partial^2}{ \partial_{z_i} \partial_{w_j} } \psi(z-w) = \frac{\partial^2}{ \partial_{z_i} \partial_{w_j} } J_0(|z-w|) \end{equation*} where |$z=(z_1,z_2)$| and |$w=(w_1,w_2)$| are two points in |$\mathbb{R}^2$|⁠. The elements of |$A$| are obtained by passing to the limit |$w\to z$|⁠. To give an example of such computation we give details of the computation of |$(A)_{1,1}$|⁠. For this we first use the chain rule to obtain \begin{align*} \frac{\partial^2}{ \partial_{z_1} \partial_{w_1} } J_0(|z-w|) &=J''_0(|z-w|)\, \left( \frac{\partial}{ \partial_{z_1} } |z-w|\right) \; \left(\frac{\partial}{ \partial_{w_1} } |z-w|\right) +J'_0( |z-w|) \, \frac{\partial^2}{ \partial_{z_1} \partial_{w_1} } |z-w|. \end{align*} The zeroth Bessel function could be defined by power series $$ J_0(x)=\sum_{n=0}^\infty \frac{(-1)^n}{(n!)^2}\left({\frac{x}{2}}\right)^{2n}. $$ From this expansion we can immediately get the expansions for |$J_0'$| and |$J_0''$| and show that $$ \lim_{w\to z} \frac{\partial^2}{ \partial_{z_1} \partial_{w_1} } J_0(|z-w|)=\frac{1}{2}. $$ In the same way we compute the other entries of |$A$| and obtain \begin{align} \label{MA} A= \left(\begin{array}{cc} \frac{1}{2} &0 \\ 0&\frac{1}{2} \end{array} \right)\!. \end{align} (B1) Since the first and second order derivatives of any stationary field are independent at every fixed point |$z \in \mathbb{R}^2$|⁠, we immediately have |$B=0$|⁠. With analogous calculations, but using higher order derivatives of |$J_0$| we compute the entries of |$C$| and find that \begin{align} \label{MC} C= \left( \begin{array}{ccc} \frac{3 }{8} &0& \frac{1}{8} \\ 0&\frac{1}{8}&0\\ \frac{1}{8}&0&\frac{3}{8} \end{array}\right)\!. \end{align} (B2) B.2: Covariance matrix of |$\boldsymbol{(\nabla \Psi(z), \nabla \Psi(w), \nabla^2 \Psi(z), \nabla^2 \Psi(w))}$| We compute the covariance matrix |${\bf \Sigma}(z,w)$| for the |$10$|-dimensional Gaussian random vector which combines the gradient and the elements of the Hessian evaluated at |$z, w$|⁠: $$(\nabla \Psi(z), \nabla \Psi(w), \nabla^2 \Psi(z), \nabla^2 \Psi(w)),$$ only for the case |$z=(0,0)$| and |$w=(0,r)$|⁠. As explained in Section 3.2 this is sufficient in order to evaluate |$K_{2}(r)$| for all relevant |$r$|⁠, thanks to the isotropic property of |$\Psi$|⁠. It is convenient to write the matrix |${\bf \Sigma}(z,w)$| in block form, that is $$ {\bf \Sigma}(z,w)={\bf \Sigma}(r)=\left(\begin{array}{ccc} {\bf A}(z,w)&{\bf B}(z,w)\\ {\bf B}^t(z,w)&{\bf C}(z,w) \end{array} \right)\!. $$ The matrix |${\bf A}$| also has a natural block structure \begin{align*} &\left. {\bf A}(z,w)\right|_{{z=(0,0),}{w=(0,r)}}={\bf A}(r)= \left(\begin{array}{cc} A & A(r) \\ A(r)&A \end{array} \right)\!, \end{align*} where |$A$| is the same as in (B1), and |$A(r)$| turns out to be a diagonal matrix, we denote its diagonal elements by |$\alpha_i(r)$| \begin{align*} &A(r)= \left(\begin{array}{cc} \alpha_1(r) & 0 \\ 0&\alpha_2(r) \end{array} \right) \end{align*} The diagonal elements |$\alpha_i$| are found by differentiating the covariance kernel of |$\Psi$|⁠: \begin{equation} \begin{aligned} \alpha_1(r)&=\left. \frac{\partial^2}{\partial_{z_1} \partial_{w_1}} J_0( |z-w|) \right|_{z=(0,0), w=(0,r)}=-J'_0( r) \frac{1}{r},\\ \alpha_2(r)&=\left. \frac{\partial^2}{\partial_{z_2} \partial_{w_2}} J_0( |z-w|) \right|_{z=(0,0), w=(0,r)}=-J''_0( r). \end{aligned} \end{equation} (B3) Again, using the block structure of |$\bf A$| we can write its determinant as $$ \text{det}({\bf A}(r))= \left({\alpha_1^2(r) - \frac{1}{4} }\right) \left({ \alpha_2^2(r) - \frac{1}{4} }\right). $$ From the Taylor series for |$J_0$| one immediately gets \begin{equation*} \begin{aligned} \alpha_1(r)&=-J'_0(r) \frac{1}{r}=\frac{1}{2}- \frac{1}{2^4} r^2+O(r^4), \\ \alpha_2(r)&=-J''_{ 0 }(r)=\frac{1}{2}- \frac{3}{2^4}r^2+O(r^4) \end{aligned} \end{equation*} so that \begin{align} \label{14:25} \det( {\bf A}(r) )=\frac{3r^4}{2^8}+O(r^6). \end{align} (B4) With analogous calculations we derive also the entries of the matrices |${\bf B}$| and |${\bf C}$|⁠: we have \begin{align*} \left. {\bf B}(z,w)\right|_{{z=(0,0),}{w=(0,r)}}={\bf B}(r)= \left( \begin{array}{cc} 0 &B(r) \\ - B(r)& 0 \end{array} \right)\!, \end{align*} where \begin{align*} B(r)=\left( \begin{array}{ccc} 0&\beta_{1}(r) &0 \\ \beta_{1}(r)&0&\beta_{2}(r) \end{array}\right) \end{align*} and, in the same way as before, we obtain explicit formulas in terms of |$J_0$| and expansions at |$r=0$| \begin{align*} \beta_1(r)& =-J''_0( r) \frac{1}{r} +J'_0( r) \frac{1}{r^2}=-\frac{r}{8}+\frac{r^3}{96}+O(r^5)\\ \beta_2(r)& = - J'''_0( r)=-\frac{3r}{8}+\frac{5r^3}{96}+O(r^5). \end{align*} In the same way \begin{align*} \left. {\bf C}(z,w)\right|_{{z=(0,0),}{w=(0,r)}}=\left( \begin{array}{cc}C & C(r) \\ C(r) &C \end{array} \right)\!, \end{align*} where |$C$| defined in (B2) and \begin{align*} C(r)= \left( \begin{array}{ccc} \gamma_{1}(r)&0&\gamma_{2}(r) \\ 0 &\gamma_{2}(r)&0 \\ \gamma_{2}(r)&0&\gamma_{3}(r) \end{array}\right)\!, \end{align*} with \begin{align*} \gamma_1(r)&= J''_0( r)\frac{3 }{r^2} - J'_0( r)\frac{3 }{r^3}=\frac{3}{8} -\frac{r^2}{32} + O(r^4), \\ \gamma_2(r)&= \frac{J'''_0( r)}{r} - J''_0( r) \frac{2 }{r^2} + J'_0( r)\frac{2 }{r^3}=\frac{1}{8} - \frac{r^2}{32} + O(r^4), \\ \gamma_3(r)& = J''''_0( r)=\frac{3}{8} - \frac{5 r^2}{32}+O(r^4). \end{align*} B.3: Conditional covariance matrix As explained before, the covariance matrix of the conditional vector $$ (\nabla^2 \Psi(z), \nabla^2 \Psi(w) | \nabla \Psi(z)= \nabla \Psi(w)=0) $$ is given by $$ {\Delta}(r)={\bf C}(r)-{\bf B}(r)^t {\bf A}(r)^{-1} {\bf B}(r)=\left( \begin{array}{cc} \Delta_1(r) & \Delta_2(r) \\ \Delta_2(r) & \Delta_1(r) \end{array} \right) $$ where |$\Delta_i$| are defined in (A1). Since we already have explicit formulas for elements of |$\bf A$|⁠, |$\bf B$|⁠, and |$\bf C$| we can obtain the following explicit formulas and expansions for |$a_i$| that define |$\Delta$|⁠: \begin{equation} \label{eq: series for a} \begin{aligned} a_1(r)&= - \frac{2 \beta_1^2(r)}{1-4 \alpha_2^2(r)}+\frac{1}{2^3 3}=- \frac{13}{2^7 3^3} r^2- \frac{151}{2^{11} 3^5} r^4- \frac{1531}{2^{15} 3^7}r^6+O(r^8),\\ a_2(r)&= -\frac{2 \beta_1^2(r)}{1-4 \alpha_1^2(r)}+\frac{1}{2^3} = \frac{1}{2^7} r^2 + \frac{1}{2^{11} 3^2} r^4 - \frac{23}{2^{15} 3^3 5} r^6+O(r^8),\\ a_3(r)&=- \frac{2 \beta_2^2(r)}{1-4 \alpha_2^2(r)}+ \frac{3}{2^3} =\frac{1}{2^7} r^2 + \frac{41}{2^{11} 3^3}r^4+ \frac{2617}{2^{15} 3^5 5 }r^6 +O(r^8),\\ a_4(r)&= - \frac{2 \beta_1(r) \beta_2(r)}{1-4 \alpha_2^2(r)}+ \frac{1}{2^3}= - \frac{5}{2^7 3^2} r^2 - \frac{23}{2^{11} 3^4} r^4 + \frac{521}{2^{15} 3^6 5} r^6 + O(r^8),\\ a_5(r)&=\gamma_1(r)- \frac{4 \alpha_2(r) \beta_1^2(r) }{1-4 \alpha_2^2(r)}- \frac{1}{3} = - \frac{67}{2^7 3^3} r^2 +\frac{7 \cdot 71}{2^{11} 3^5}r^4 +\frac{13 \cdot 271}{2^{15} 3^7 5}r^6 +O(r^8),\\ a_6(r)&= \gamma_2(r) - \frac{4 \alpha_1(r) \beta_1^2(r)}{1-4 \alpha_1^2(r)}= -\frac{1}{2^7} r^2 + \frac{1}{2^{11} 3^2} r^4 +\frac{19}{2^{15} 3^3 5}r^6 +O(r^8), \\ a_7(r)&= \gamma_3(r)- \frac{4 \alpha_2(r) \beta_2^2(r)}{1-4 \alpha_2^2(r)} = -\frac{1}{2^7}r^2 -\frac{31}{2^{11} 3^3} r^4 -\frac{2621}{2^{15} 3^5 5} r^6 +O(r^8),\\ a_8(r)&= \gamma_2(r) - \frac{4 \alpha_2(r) \beta_1(r) \beta_2(r)}{1-4 \alpha_2^2(r)}= \frac{13}{2^7 3^2}r^2 -\frac{23}{2^{11} 3^4}r^4 + \frac{7}{2^{15} 3^6}r^6 +O(r^8). \end{aligned} \end{equation} (B5) Communicated by Prof. Misha Sodin References [1] Adler R. and Taylor. 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Kasjan,, Stanisław;Keller,, Gerhard;Lemańczyk,, Mariusz

doi: 10.1093/imrn/rnx196pmid: N/A

Abstract Let |${\mathcal B}$| be an infinite subset of |$\{1,2,\dots\}$|⁠. We characterize arithmetic and dynamical properties of the |${\mathcal B}$|-free set |${\mathcal F}_{\mathcal B}$| through group theoretical, topological and measure theoretic properties of a set |$W$| (called the window) associated with |${\mathcal B}$|⁠. This point of view stems from the interpretation of the set |${\mathcal F}_{\mathcal B}$| as a weak model set. Our main results are: |${\mathcal B}$| is taut if and only if the window is Haar regular; the dynamical system associated to |${\mathcal F}_{\mathcal B}$| is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to |${\mathcal F}_{\mathcal B}$| is proximal if and only if the window has empty interior; and the dynamical system associated to |${\mathcal F}_{\mathcal B}$| has the “naïvely expected” maximal equicontinuous factor if and only if the interior of the window is aperiodic. 1 Introduction and Main Results For any given set |${\mathcal B}\subseteq{\mathbb N}=\{1,2,\dots\}$| one can define its set of multiples \begin{equation*} {\mathcal M}_{\mathcal B}:=\bigcup_{b\in{\mathcal B}}b{\mathbb Z} \end{equation*} and the set of |${\mathcal B}$|-free numbers \begin{equation*} {\mathcal F}_{\mathcal B}:={\mathbb Z}\setminus{\mathcal M}_{\mathcal B}. \end{equation*} The investigation of structural properties of |${\mathcal M}_{\mathcal B}$| or, equivalently, of |${\mathcal F}_{\mathcal B}$| has a long history (see the monograph [10] and the recent article [4] for references), and dynamical systems theory provides some useful tools for this. Namely, denote by |$\eta\in\{0,1\}^{\mathbb Z}$| the characteristic function of |${\mathcal F}_{\mathcal B}$|⁠, that is |$\eta(n)=1$| if and only if |$n\in{\mathcal F}_{\mathcal B}$|⁠, and consider the orbit closure |$X_\eta$| of |$\eta$| in the shift dynamical system |$(\{0,1\}^{\mathbb Z},\sigma)$|⁠, where |$\sigma$| stands for the left shift. Then topological dynamics and ergodic theory provide a wealth of concepts to describe various aspects of the structure of |$\eta$|⁠, see [16] which originated this point of view by studying the set of square-free numbers, and also [1, 4], which continued this line of research. In this article we continue to provide a dictionary that characterizes arithmetic properties of |${\mathcal B}$| in terms of dynamical properties of |$X_\eta$|⁠, and, as an intermediate step, also in terms of topological and measure theoretic properties of a pair |$(H,W)$| associated with the passage from |${\mathcal B}$| to |$X_\eta$|⁠, where |$H$| is a compact abelian group and |$W$| a compact subset of |$H$|⁠. This latter point of view is borrowed from the theory of weak model sets, which applies here, because |${\mathcal F}_{\mathcal B}$| is a particular example of such a set, see for example, [3, 13]. Finally the Chinese Remainder Theorem allows us to interpret our dynamical results combinatorially. In order to formulate our main results, we need to recall some notions from the theory of sets of multiples [10] and also to introduce some further notation. Let |${\mathcal B}$| be a non-empty subset of |${\mathbb N}$|⁠. |${\mathcal B}$| is primitive, if there are no different |$b,b'\in{\mathcal B}$| with |$b\mid b'$|⁠. From any set |${\mathcal B}\subseteq{\mathbb N}$| one can remove all multiples of other numbers in |${\mathcal B}$|⁠, which results in the set \begin{equation}\label{eq:Bprim} {{\mathcal B}}^{prim}:={\mathcal B}\setminus \bigcup_{b\in{\mathcal B}}b\cdot({\mathbb N}\setminus\{1\}). \end{equation} (1)|${{\mathcal B}}^{prim}$| is primitive by construction, and |${\mathcal M}_{\mathcal B}={\mathcal M}_{{{\mathcal B}}^{prim}}$|⁠. |${\mathcal B}$| is taut, if |${\boldsymbol{\delta}}({\mathcal M}_{{\mathcal B}\setminus\{b\}})<{\boldsymbol{\delta}}({\mathcal M}_{\mathcal B})$| for each |$b\in{\mathcal B}$|⁠, where |${\boldsymbol{\delta}}({\mathcal M}_{\mathcal B})\,{:=}\lim_{n\to\infty}\frac{1}{\log n}\sum_{k\leqslant n,k\in{\mathcal M}_{\mathcal B}}k^{-1}$| denotes the logarithmic density of this set, which is known to exist by the theorem of Davenport and Erdös [6, 7]. So a set is taut, if removing any single point from it changes its set of multiples drastically and not only by “a few points”. |$\tilde{H}:=\prod_{b\in{\mathcal B}}{\mathbb Z}/b{\mathbb Z}$| and |$\Delta:{\mathbb Z}\to\tilde{H}$|⁠, |$\Delta(n)=(n \mod b)_{b\in{\mathcal B}}$| – the canonical diagonal embedding. |$H:=\overline{\Delta({\mathbb Z})}$| is a compact abelian group, and we denote by |$m_H$| its normalized Haar measure. |$R_{\Delta(1)}: H\to H$| denotes the rotation by |$\Delta(1)$|⁠, that is |$(R_{\Delta(1)}h)_b=(h_b+1)$| mod |$b$| for all |$b\in{\mathcal B}$|⁠. As |$\{R_{\Delta(1)}^n(\Delta(0)):n\in{\mathbb Z}\}=\Delta({\mathbb Z})$| is dense in |$H$|⁠, the system |$(H,R_{\Delta(1)})$| is strictly ergodic. The window is defined as \begin{equation}\label{eq:W} W:=\{h\in H: h_b\neq0\ (\forall b\in{\mathcal B})\}. \end{equation} (2) |$\varphi:H\to\{0,1\}^{\mathbb Z}$| is the coding function: |$\varphi(h)(n)=1$|⁠, if and only if |$R_{\Delta(1)}^nh\in W$|⁠, equivalently, if and only if |$h_b+n\neq0$| mod |$b$| for all |$b\in{\mathcal B}$|⁠. By |$S,S'\subset{\mathcal B}$| we always mean finite subsets. The topology on |$H$| is generated by the (open and closed) cylinder sets \begin{equation*} U_S(h):=\{h'\in H: \forall b\in S: h_b=h'_b\},\text{ defined for finite }S\subset{\mathcal B}\text{ and }h\in H. \end{equation*} A recurring theme of the main results in this paper is to characterize arithmetic and dynamical properties of a |${\mathcal B}$|-free set |${\mathcal F}_{\mathcal B}$| through group theoretical, topological and measure theoretic properties of the window |$W$| defined above. Remark 1.1. With the notation introduced above, we can write \begin{equation*} X_\eta=\overline{\varphi(\Delta({\mathbb Z}))}. \end{equation*} This is certainly a subset of |$X_\varphi:=\overline{\varphi(H)}$|⁠, the set studied in [13] under the name |${{\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}}$|⁠. In Proposition 2.2 we show that |$X_\eta=X_\varphi$| when |${\mathcal B}$| has light tails (see Subsection 2.5 for a definition), but we do not know whether also tautness of |${\mathcal B}$| suffices (see also Subsection 2.5). □ The authors wish to thank the Referee for valuable remarks and also Aurelia Bartnicka for reading the text and suggesting several improvements. 1.1 Tautness as a measure theoretic property Theorem A. Suppose that the set |${\mathcal B}$| is primitive. Then the following are equivalent: (i) |${\mathcal B}$| is taut. (ii) The window |$W$| is Haar regular, that is |${\mathrm{supp}}(m_H|_W)=W$|⁠. Moreover, these properties imply (iii) |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠. □ The proof of the theorem is provided in Section 2. The concept of a Haar regular window was introduced in [14] in the context of general weak model sets. (The authors are indebted to J. Kułaga-Przymus for pointing out the relevance of [10, Lemma 1.17] for the proof of this theorem.) Given a set |${\mathcal B}\subset{\mathbb N}$|⁠, one says that |$h:=(h_b)_{b\in{\mathcal B}}\in{\mathbb Z}^{{\mathcal B}}$| satisfies the CRT (Chinese Remainder Theorem) if for each finite |$S\subset{\mathcal B}$| there exists |$n\in{\mathbb Z}$| such that, \begin{equation}\label{freeCRT} h_b=n\text{ mod }b\;\text{ for each }b\in S. \end{equation} (3) Clearly, \begin{equation*} h\,\,\text{satisfies the CRT iff}\,\,h\in H. \end{equation*} We are looking for solutions of (3) with |$n\in{\mathcal F}_{\mathcal B}$|⁠. If for |$h$| as above we can solve (3) with |$n=n_S\in{\mathcal F}_{\mathcal B}$| for all finite |$S\subset {\mathcal B}$|⁠, then we say that |$h$| satisfies the |${\mathcal B}$|-free CRT. A necessary condition for |$h=(h_b)_{b\in{\mathcal B}}$| to satisfy the |${\mathcal B}$|-free CRT is, of course, that |$h_b\,{\neq}\,0\mod b$| for each |$b\in{\mathcal B}$|⁠, and a moment’s reflection shows that \begin{equation}\label{eq:B-free-CRT} h\text{ satisfies the}\,{\mathcal B}\,{\text{-free CRT iff}}\,h\in \overline{\Delta({\mathbb Z})\cap W}. \end{equation} (4) Therefore the implication |$(i)$||$\Rightarrow$||$(iii)$| of Theorem A is an immediate consequence of the following proposition. Proposition 1.1. Assume that |${\mathcal B}$| is taut. Let |$h\in W$| and |$S\subset{\mathcal B}$| finite. Then the set of |${\mathcal B}$|-free integers |$n$| that solve |$n = h_b$| mod |$b$| for |$b\in S$| has asymptotic density |$m_H(U_S(h)\cap W)\,{>}\,0$|⁠. □ In Subsection 2.4 we provide a sequence |${\mathcal B}$|⁠, which is not taut, but for which |$\overline{\Delta({\mathbb Z})\cap W}=W$| (Example 2.2). Hence (iii) of Theorem A is not equivalent to (i) and (ii). Here we provide two simpler examples which throw some light on property (iii). Denote by |${\mathcal P}\subseteq{\mathbb N}$| the set of all prime numbers. Example 1.1. If |${\mathcal B}={\mathcal P}$| then |$H=\prod_{p\in{\mathcal P}}{\mathbb Z}/p{\mathbb Z}$|⁠, |$W$| is uncountable (although of Haar measure zero) and |$\overline{\Delta({\mathbb Z})\cap W}\neq W$|⁠, since for each |$n$| we find |$p\in{\mathcal P}$| such that |$p\mid n$|⁠, so |$n=0$| mod |$p$|⁠. □ Example 1.2. If |${\mathcal B}\subset{{\mathcal P}}$| is thin, that is if |$\sum_{p\in{\mathcal B}}1/p<+\infty$|⁠, then |$\overline{\Delta({\mathbb Z})\cap W}=W$| in view of (4), because each |$h\in H$| satisfies the |${\mathcal B}$|-free CRT. Indeed, if |$S\subset{\mathcal B}$| is finite and |$n=h_b\mod b$| for |$b\in S$|⁠, then |$n+{\mathrm{lcm}}(S){\mathbb Z}$| is the set of all solutions to this system of congruences. Moreover, if |$h\in W$|⁠, then |$\gcd(n,b)=1$| for all |$b\in S$|⁠. We only need to find |$r\in{\mathbb Z}$| so that |$n+r{\mathrm{lcm}}(S)$| is a prime number which is not in |${\mathcal B}$|⁠. The latter follows from Dirichlet’s theorem: The set of prime numbers contained in |$n+{\mathrm{lcm}}(S){\mathbb Z}$| is not thin. Of course this is a special case of Theorem A. □ Remark 1.2. Denote by |$\nu_\eta:=m_H\circ\varphi^{-1}$| the Mirsky measure on |$X_\eta$|⁠. There are two independent proofs of the fact that the two equivalent conditions from Theorem A imply that the measure preserving dynamical system |$(X_\eta,\sigma,\nu_\eta)$| is isomorphic to the group rotation |$(H,R_{\Delta(1)},m_H)$|⁠: In [4, Theorem F] it is proved that this is implied by |$(i)$|⁠. That it is also a direct consequence of |$(ii)$| follows from [14] (in the more general context of model sets). The proof uses our observation that |$W$| is aperiodic (see Proposition 5.1). To see this, denote by |$H_W:=\{h\in H: W+h=W\}$| the period group of |$W$| and by |$H_W^{Haar}:=\{h\in H: m_H((W+h)\triangle W)=0\}$| its group of Haar periods. It is easily seen that |$H_W=H_W^{Haar}$| for Haar regular |$W$|⁠, in particular whenever the sequence |${\mathcal B}$| is taut. Hence, if |$W$| is aperiodic, it is also Haar aperiodic, and this is what is needed to apply the general theorem from [14] to the present context. A word of caution is in order at this point: Although, in the |${\mathcal B}$|-free context, the window |$W$| is always aperiodic (Proposition 5.1), this is not necessarily true for its Haar regularization |$W_{reg}:={\mathrm{supp}}(m_H|_W)$|⁠, because that window is not of the same arithmetic type as |$W$|⁠. On the other hand, as proved in [4, Theorem C], each non-taut set |${\mathcal B}$| can be modified into a taut set |${\mathcal B}'$| whose corresponding Mirsky measure |$\nu_{\eta'}$| coincides with |$\nu_\eta$| (as a measure on |$\{0,1\}^{\mathbb Z}$|⁠). The (arithmetic!) window |$W'\subseteq H'$| defined by |${\mathcal B}'$| is then aperiodic and Haar regular, and we suspect that it to be closely related to |$W_{reg}\subseteq H$|⁠. □ 1.2 The proximal and the Toeplitz case From [4, Theorem A] we know that |$X_\eta$| has a unique minimal subset |$M$|⁠. In Lemma 3.10, we prove that |$M=\overline{\varphi(C_\varphi)}$|⁠, where |$C_\varphi$| denotes the set of continuity points of |$\varphi:H\to\{0,1\}^{\mathbb Z}$|⁠, see also [13, Lemma 6.3]. |$M$| is degenerate to a singleton, namely to |$M= \{(\dots,0,0,0,\dots)\}$|⁠, if and only if |${\mathrm{int}}(W)=\emptyset$| [13], and we collect a number of equivalent characterizations of this extreme case in Theorem C below. Assuming primitivity of |${\mathcal B}$| and property |$(iii)$| of Theorem A, we prove the following equivalent characterizations of minimality of |$(X_\eta,\sigma)$|⁠, that is of |$M=X_\eta$|⁠, in Subsection 3.2. For |$S\subset{\mathcal B}$| let \begin{equation}\label{eq:A_S-def} {\mathcal A}_S:=\{\gcd(b,{\mathrm{lcm}}(S)): b\in{\mathcal B}\}, \end{equation} (5) and note that |${\mathcal F}_{{\mathcal A}_S}\subseteq{\mathcal F}_{\mathcal B}$|⁠, because |$b\mid m$| for some |$b\in{\mathcal B}$| implies |$\gcd(b,{\mathrm{lcm}}(S))\mid m$| for any |$S\subset{\mathcal B}$|⁠. Let \begin{equation}\label{def:Cinf} {{\mathcal A}_\infty}:=\{n\in{\mathbb N}: \forall_{S\subset{\mathcal B}}\ \exists_{S': S\subseteq S'{\subset{\mathcal B}}}: n\in{\mathcal A}_{S'}\setminus S'\}. \end{equation} (6) In Lemma 3.2 we prove: If |$(S_k)_k$| is a filtration of |${\mathcal B}$| with finite sets, then \begin{equation} \limsup_{k\to\infty}\left({\mathcal A}_{S_k}\setminus S_k\right)={{\mathcal A}_\infty}. \end{equation} (7) Theorem B. Suppose that |${\mathcal B}$| is primitive. Consider the following list of properties: (B1) The window |$W$| is topologically regular, that is |$\overline{{\mathrm{int}}(W)}=W$|⁠. (B2) |${\mathcal F}_{\mathcal B}=\bigcup_{S\subset{\mathcal B}\text{ finite}}{\mathcal F}_{{\mathcal A}_S}$|⁠. (B3) |${{\mathcal A}_\infty}=\emptyset$|⁠. (B4) There are no |$d\in{\mathbb N}$| and no infinite pairwise coprime set |${\mathcal A}\subseteq{\mathbb N}\setminus\{1\}$| such that |$d\,{\mathcal A}\subseteq{\mathcal B}$|⁠. (B5) |$\eta=\varphi({\Delta(0)})$| is a Toeplitz sequence (see [8], [12] for the definition) different from |$(\dots,0,0,0,\dots)$|⁠. (B6) |$0\in C_\varphi$| and |$\varphi(0)\neq(\dots,0,0,0,\dots)$|⁠. (B7) |$\eta\in M$| and |$\eta\neq(\dots,0,0,0,\dots)$|⁠. (B8) |$X_\eta$| is minimal., that is |$X_\eta=M$|⁠, and |${\mathrm{card}}(X_\eta)>1$|⁠. (B9) The dynamics on |$X_\eta$| is a minimal almost 1-1 extension of |$(H,R_{\Delta(1)})$|⁠, the rotation by |$\Delta(1)$| on |$H$|⁠. (a) (B1)–(B6) are all equivalent, and each of these conditions implies that |${\mathcal B}$| is taut. (b) (B7) and (B8) are equivalent. (c) Each of (B1)–(B6) implies (B9). (d) (B9) implies (B7) and (B8). (e) If |$\overline{\Delta({\mathbb Z})\cap W}=W$| (in particular if |${\mathcal B}$| is taut), then (B1)–(B9) are all equivalent. □ Remark 1.3. One ingredient of the proof of Theorem B is the observation that the set |${\mathcal B}$| is taut whenever |$\eta$| is a Toeplitz sequence. This was pointed out to us by A. Bartnicka who also gave a proof of it, which we recall in Lemma 3.7 below. Moreover, we can interpret the result purely arithmetically as follows: If |${\mathcal B}$| is primitive and satisfies (B4) then the set of elements for which the |${\mathcal B}$|-free CRT holds is topologically regular, that is it contains a dense subset of points for which all sufficiently close points satisfying the CRT satisfy also the |${\mathcal B}$|-free CRT. □ The following characterization of regular Toeplitz sequences is included in Proposition 4.1 in Subsection 4.2, where also the precise definition of regularity of a Toeplitz sequence is recalled. Proposition 1.2. Assume that |$\overline{{\mathrm{int}}(W)}=W$|⁠. Then the Toeplitz sequence |$\eta$| is regular, if and only if |$m_H(\partial W)=0$|⁠. □ In Subsection 4.2 we also provide examples of sets |${\mathcal B}$| that give rise to regular Toeplitz sequences and others giving rise to irregular Toeplitz sequences. In Lemma 4.3 we prove that |$m_H(\partial W)=0$| if and only if |$\inf_{S\subset{\mathcal B}}\bar d({\mathcal M}_{{\mathcal A}_S}\setminus{\mathcal M}_{\mathcal B})=0$|⁠, where |$\bar d({\mathcal C})$| denotes the upper density of a set |${\mathcal C}\subseteq{\mathbb N}$|⁠, see Definition 2.1. In this context one may note that |$m_H(\partial W)=0$| implies unique ergodicity of the dynamics on |$X_\eta$| [13, Theorem 2c]. The next theorem is complementary to Theorem B. Most of its equivalences follow from results in [4] and [13] and are proved in Subsection 3.3. They do not rely on the more advanced arithmetic concept of tautness. Theorem C. The following are equivalent: (C1) |${\mathrm{int}}(W)=\emptyset$|⁠. (C2) |$\bigcup_{S\subset{\mathcal B}\text{ finite}}{\mathcal F}_{{\mathcal A}_S}=\emptyset$|⁠, that is |${\mathcal F}_{{\mathcal A}_S}=\emptyset$| for all finite |$S\subset{\mathcal B}$|⁠. (C3) |$\forall S\subset{\mathcal B}: 1\in{\mathcal A}_S$|⁠. (C4) |${\mathcal B}$| contains an infinite pairwise coprime subset. (C5) If |${\mathcal C}\subseteq{\mathbb N}$| is finite and if |${\mathcal B}\subseteq{\mathcal M}_{\mathcal C}$|⁠, then |$1\in{\mathcal C}$|⁠. (C6) |$M=\{(\dots,0,0,0,\dots)\}$|⁠. (C7) The dynamics on |$X_\eta$| are proximal. □ Remark 1.4. Under the conditions of Theorem C no element of |$W$| is stable, that is, for each |$h$| satisfying the |${\mathcal B}$|-free CRT there is an element |$h'\in{\mathbb Z}^{{\mathcal B}}$| arbitrarily close to |$h$| which satisfies the CRT but not the |${\mathcal B}$|-free CRT. □ 1.3 The maximal equicontinuous factor We finish with a result that identifies the maximal equicontinuous factor of the dynamics on |$X_\eta$| and answers Question 3.14 in [4]. Given a subset |$A\subseteq H$|⁠, denote by \begin{equation*} H_A:=\left\{h\in H: A+h=A\right\} \end{equation*} the period group of |$A$|⁠. The set |$A\subseteq H$| is topologically aperiodic, if |$H_A=\{0\}$|⁠. Observe also that |$H_{{\mathrm{int}}(A)}$| is a closed subgroup of |$H$|⁠, whenever |$A$| is closed [14, Lemma 6.1]. In Proposition 5.1 we prove that |$H_W=\{0\}$| whenever |${\mathcal B}$| is primitive. If |${\mathrm{int}}(W)=\emptyset$|⁠, then of course |$H_{{\mathrm{int}}(W)}=H$|⁠. If |${\mathrm{int}}(W)\neq\emptyset$|⁠, the situation is more complicated: |$H_{{\mathrm{int}}(W)}$| is obviously always a strict subgroup of |$H$|⁠, and very often |$H_{{\mathrm{int}}(W)}=\{0\}$|⁠, but there are examples where |$H_{{\mathrm{int}}(W)}$| is a non-trivial strict subgroup of |$H$|⁠, see Subsection 5.3. In any case, however, |$H_{{\mathrm{int}}(W)}$| determines the maximal equicontinuous factor. The following is proved in [14, Theorem A2]: \begin{equation} \textbf{Theorem}\; {\it{The\,\,translation\,\,by}}\,\,\Delta(1)+H_{{\mathrm{int}}(W)}\,\,{\it{on}}\,\,H/H_{{\mathrm{int}}(W)}\,{\it{is\,\,the\,\,maximal}}\\ {\it{equicontinuous\,\,factor\,\,of\,\,the\,\,dynamics\,\,on}} X_\eta. \end{equation} (8) Let |$S_1\subset S_2\subset\dots$| be any filtration of |${\mathcal B}$| by finite sets. In Subsection 5.1 we define divisors |$d_k$| of |${\mathrm{lcm}}(S_k)$|⁠: \begin{equation} d_k:=\lim_{j\to\infty}\gcd(s_k,c_{k+j})\,,\;\text{ where }\; s_k:={\mathrm{lcm}}(S_k)\;\text{ and }\; c_l:=\text{minimal period of }{\mathcal M}_{{\mathcal A}_{S_l}}, \end{equation} (9) which means that |$c_\ell$| is the minimal positive integer such that |${\mathcal M}_{{\mathcal A}_{S_l}}+c_\ell={\mathcal M}_{{\mathcal A}_{S_l}}$|⁠. By Remark 5.1 we have |$\frac{s_k}{d_k}\mid\frac{s_{k+1}}{d_{k+1}}$| for any |$k$|⁠. The sequences |$(s_k)$|⁠, |$(d_k)$| and |$(c_k)$| determine |$H_{{\mathrm{int}}(W)}$| in the following way: Proposition 1.3. (a) |$0\rightarrow H_{{\mathrm{int}}(W)}\rightarrow H\cong\lim\limits_{\leftarrow}{\mathbb Z}/{s_k}{\mathbb Z}\rightarrow \lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}\rightarrow 0$| is an exact sequence. Recall that a sequence of abelian groups and homomorphisms |$...\ {\smash{\mathop{\longrightarrow}\limits^{{}}_{{}}}} M_{k-1}\smash{\mathop{\longrightarrow}\limits^{{f_{k-1}}}_{{}}} M_{k}\smash{\mathop{\longrightarrow}\limits^{{f_k}}_{{}}} M_{k+1}\rightarrow...$| is called exact if the kernel of |$f_k$| is equal to the image of |$f_{k-1}$| for any |$k$|⁠. In particular, a sequence $$ 0\rightarrow M' \smash{\mathop{\longrightarrow}\limits^{{f}}_{{}}} M\smash{\mathop{\longrightarrow}\limits^{{g}}_{{}}} M''\rightarrow 0 $$ is exact, when |$f$| is injective, the kernel of |$g$| equals the image of |$f$| and |$g$| is surjective. We say that it is a “short exact sequence”. In particular, the homomorphism |$g$| induces an isomorphism |$M''\cong M/f(M')$| in this case. (b) |$H_{{\mathrm{int}}(W)}\cong \lim\limits_{\leftarrow}{\mathbb Z}/\frac{s_k}{d_k}{\mathbb Z}$|⁠. (c) |$H/H_{{\mathrm{int}}(W)}\cong \lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}$|⁠. (d) |$H_{{\mathrm{int}}(W)}=\{0\}$| if and only if |$s_k=d_k$| for each |$k\in{\mathbb N}$|⁠, equivalently if for each |$b\in{\mathcal B}$| there is |$n>0$| such that |$b$| divides |$c_{n}$|⁠. □ Theorem D. (a) The translation by |$(1,1,\dots)$| on |$H/H_{{\mathrm{int}}(W)}\cong\lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}$| is the maximal equicontinuous factor of the dynamics on |$X_\eta$|⁠. (b) In case (d) of Proposition 1.3, the translation by |$\Delta(1)$| on |$H\cong \lim\limits_{\leftarrow}{\mathbb Z}/{s_k}{\mathbb Z}$| is the maximal equicontinuous factor of the dynamics on |$X_\eta$|⁠. □ In Subsection 5.3 we provide a number of examples illustrating this theorem. Remark 1.5. In [4], the following set |$Y$| is defined, versions of which occur also in [15] and [2]: \begin{equation*} Y:=\left\{x\in\{0,1\}^{\mathbb Z}: {\mathrm{card}}({\mathrm{supp}}(x)\mod b)=b-1\;\forall b\in{\mathcal B}\right\}\!. \end{equation*} Observe that |${\mathrm{card}}({\mathrm{supp}}(x)\mod b)\leqslant b-1$| for all |$x\in X_\eta$| and |$b\in{\mathcal B}$|⁠. Indeed, if |${\mathrm{card}}({\mathrm{supp}}(x)\mod b)= b$| for some |$x\in X_\eta$| and |$b\in{\mathcal B}$|⁠, then this happens on some integer interval |$[-M,M]$|⁠, and hence |${\mathrm{card}}({\mathrm{supp}}(\eta)\mod b)= b$|⁠, which contradicts the fact that |${\mathrm{supp}}(\eta)\subseteq{\mathcal F}_{\mathcal B}$|⁠. Proposition 3.27 of [4] asserts that |$(H,R_{\Delta(1)})$| is the maximal equicontinuous factor of |$(X_\eta,S)$|⁠, whenever |$X_\eta\subseteq Y$|⁠. Hence, in that case, |$H_{\overline{{\mathrm{int}} W}}= H_{{\mathrm{int}} W}=\{0\}=H_W$| by Theorem D and Proposition 5.1. This is the second one of the following two implications: \begin{align} W=\overline{{\mathrm{int}} W}\quad\Rightarrow\quad X_\eta\subseteq Y\quad\Rightarrow\quad H_W=H_{\overline{{\mathrm{int}} W}}. \end{align} (10) The first one is proved in Proposition 3.3. □ 2 Tautness of |$\boldsymbol{{\mathcal B}}$| and Haar Regularity of |$\boldsymbol{W}$| 2.1 Arithmetic of |${\mathcal B}$| and topology of |$W$|⁠, part I Definition 2.1. Let |${\mathcal M}\subseteq{\mathbb N}$|⁠. (a) The upper resp. lower density of |${\mathcal M}$| is \begin{align*} \overline{d}({\mathcal M})& =\limsup_{N\to\infty} \frac1N{\mathrm{card}}\left({\mathcal M}\cap\{1,\dots,N\}\right) \text{ resp. }\\[-2pt] \underline{d}({\mathcal M})&=\liminf_{N\to\infty} \frac1N{\mathrm{card}}\left({\mathcal M}\cap\{1,\dots,N\}\right). \end{align*} If the limit exists, we write |$d({\mathcal M})$|⁠. (b) The logarithmic density of |${\mathcal M}$| is \begin{equation*} {\boldsymbol{\delta}}({\mathcal M})=\lim_{N\to\infty}\frac{1}{\log N}\sum_{n\in{\mathcal M}\cap\{1,\dots,N\}}\frac{1}{n} \end{equation*} whenever the limit exists. □ The theorem of Davenport and Erdös [6, 7] asserts that |${\boldsymbol{\delta}}({\mathcal M}_{\mathcal B})=\underline{d}({\mathcal M}_{\mathcal B})$| exists for any subset |${\mathcal B}\subseteq{\mathbb N}$|⁠. Definition 2.2. |${\mathcal B}\subseteq{\mathbb N}\setminus\{1\}$| is a Behrend sequence, if |${\boldsymbol{\delta}}({\mathcal M}_{\mathcal B})=1$|⁠. □ Recall that |${\mathcal B}$| is {taut}, if |${\boldsymbol{\delta}}({\mathcal M}_{{\mathcal B}\setminus\{b\}})<{\boldsymbol{\delta}}({\mathcal M}_{\mathcal B})$| for each |$b\in{\mathcal B}$|⁠. The following is a corollary to a theorem of Behrend [5]: Proposition 2.1. A set |${\mathcal B}\subseteq{\mathbb N}$| is taut, if and only if it is primitive and there are no |$q\in{\mathbb N}$| and no Behrend set |${\mathcal A}\subseteq {\mathbb N}\setminus\{1\}$| such that |$q\,{\mathcal A}\subseteq{\mathcal B}$| [10, Corollary 0.19]. □ This motivates the next definition: Definition 2.3. A set |${\mathcal B}\subseteq{\mathbb N}$| is pre-taut, if there are no |$ q\in{\mathbb N}$| and Behrend set |${\mathcal A}\subseteq{\mathbb N}\setminus\{1\}$| such that |$q\,{\mathcal A}\subseteq{\mathcal B}$|⁠. □ Remark 2.1. |${\mathcal B}$| is taut if and only if it is pre-taut and primitive. □ Lemma 2.1. Let |${\mathcal B}\subseteq{\mathbb N}$| and |$c\in{\mathbb N}$|⁠. (a) If |$c\,{\mathcal B}$| is pre-taut, then also |${\mathcal B}$| is pre-taut. Moreover, |${\mathcal B}$| is taut if and only if |$c\,{\mathcal B}$| is taut. (b) Each subset of a (pre-)taut set is (pre-)taut. (c) A finite union of pre-taut sets is pre-taut. (d) If |${\mathcal B}$| is taut, then |${\mathcal B}=\{1\}$| or |$d({\mathcal M}_{\mathcal B})\neq1$| (possibly non-existing). Equivalently, if |$d({\mathcal M}_{\mathcal B})=1$|⁠, then |${\mathcal B}=\{1\}$| or |${\mathcal B}$| is not taut. (e) If |${\mathcal B}$| is pre-taut, then |$1\in{\mathcal B}$| or |$d({\mathcal M}_{\mathcal B})\neq1$| (possibly non-existing). Equivalently, if |$d({\mathcal M}_{\mathcal B})=1$|⁠, then |$1\in{\mathcal B}$| or |${\mathcal B}$| is not pre-taut. □ Proof. (a) The first implication is obvious. It is also clear that |${\mathcal B}$| is primitive if and only if |$c{\mathcal B}$| is primitive. Moreover, \begin{equation*} \begin{split} {\mathcal B}\text{ is taut} &\Leftrightarrow \forall b\in{\mathcal B}: \underline{d}({\mathcal M}_{\mathcal B})>\underline{d}({\mathcal M}_{{\mathcal B}\setminus\{b\}}) \Leftrightarrow \forall b\in{\mathcal B}: c^{-1}\underline{d}({\mathcal M}_{\mathcal B})>c^{-1}\underline{d}({\mathcal M}_{{\mathcal B}\setminus\{b\}})\\ &\Leftrightarrow \forall b\in{\mathcal B}: \underline{d}({\mathcal M}_{c{\mathcal B}})>\underline{d}({\mathcal M}_{c({\mathcal B}\setminus\{b\})}) =\underline{d}({\mathcal M}_{c\,{\mathcal B}\setminus\{cb\}})\\ &\Leftrightarrow \forall b'\in c\,{\mathcal B}: \underline{d}({\mathcal M}_{c{\mathcal B}})>\underline{d}({\mathcal M}_{c\,{\mathcal B}\setminus\{b'\}})\\ &\Leftrightarrow c\,{\mathcal B}\text{ is taut.} \end{split} \end{equation*} (b) is obvious (see Remark 2.1). (c) follows from [10, Corollary 0.14], see also [4, Proposition 2.33]. (d) Suppose that |${\mathcal B}$| is taut. Then |${\mathcal B}$| is primitive, and |$d({\mathcal M}_{\mathcal B})\neq1$| unless |$1\in{\mathcal B}$| by [10, Corollary 0.19]. Hence |$d({\mathcal M}_{\mathcal B})\neq1$| or |${\mathcal B}=\{1\}$|⁠. (e) follows directly from Definition 2.3 (note that (d) follows from (e) and Remark 2.2). ■ Remark 2.2. If |${\mathcal B}$| is pre-taut, then |${{\mathcal B}}^{prim}$| is taut in view of Lemma 2.1b. □ For |$q\in{\mathbb N}$| and |${\mathcal B}\subseteq{\mathbb Z}$| let \begin{equation*} {\mathcal B}'(q)=\left\{\frac{b}{\gcd(b,q)}:b\in{\mathcal B}\right\}\!, \end{equation*} and note that |$1\in{\mathcal B}'(q)$| if and only if |$q\in{\mathcal M}_{\mathcal B}$|⁠. Lemma 2.2. Let |$q\in{\mathbb N}$|⁠, |${\mathcal B},{\mathcal C}\subseteq{\mathbb Z}$|⁠, and |$q\,{\mathcal C}\subseteq{\mathcal M}_{\mathcal B}$|⁠. Then |${\mathcal M}_{\mathcal C}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$|⁠. □ Proof. Let |$c\in {\mathcal C}$|⁠. There are |$\ell\in{\mathbb Z}$| and |$b\in{\mathcal B}$| such that |$qc=\ell b$|⁠. Since |$q\mid\ell b$|⁠, it follows that |$q\mid\ell \gcd(b,q)$|⁠, thus |$k=\frac{\ell \gcd(b,q)}{q}$| is an integer. We have \begin{equation*} c= \frac{\ell b}{q} = k\cdot\frac{b}{\gcd(b,q)} \in{\mathcal M}_{{\mathcal B}'(q)}. \end{equation*} This shows that |${\mathcal C}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$| and hence also |${\mathcal M}_{\mathcal C}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$|⁠. ■ Lemma 2.3. Let |${\mathcal B}\subseteq{\mathbb N}$| and |$q\in{\mathbb N}$|⁠. (a) If |${\mathcal B}$| is pre-taut, then |${\mathcal B}'(q)$| is pre-taut. (b) If |${\mathcal B}$| is taut, then |${\mathcal B}'(q)$| is a finite disjoint union of taut sets |${\mathcal B}_i'$| defined below in the proof of a). (c) If |$d({\mathcal M}_{\mathcal B})=1$|⁠, then |$d({\mathcal M}_{{\mathcal B}'(q)})=1$|⁠. (d) If |${\mathcal B}=\bigcup_{i=1}^N{\mathcal C}_i$| and if |$d({\mathcal M}_{\mathcal B})=1$|⁠, then |$d({\mathcal M}_{{\mathcal C}_i})=1$| for at least one |$i\in\{1,\dots,N\}$|⁠. □ Proof. Let |$I:=\left\{\frac{q}{\gcd(b,q)}:b\in{\mathcal B}\right\}$|⁠. For |$i\in I$| denote |${\mathcal B}_i:=\left\{b\in{\mathcal B}:\frac{q}{\gcd(b,q)}=i\right\}$| and |${\mathcal B}_i':=\left\{\frac{b}{\gcd(b,q)}: b\in{\mathcal B}_i\right\}$|⁠. Then |$I$| is finite, |${\mathcal B}=\bigcup_{i\in I}{\mathcal B}_i$|⁠, and |${\mathcal B}'(q)=\bigcup_{i\in I}{\mathcal B}_i'$|⁠. Moreover, |${\mathcal B}_i=\left\{\frac{q}{i}b':b'\in{\mathcal B}_i'\right\}=\frac{q}{i}{\mathcal B}_i'$|⁠. (a) If |${\mathcal B}$| is pre-taut, then all |${\mathcal B}_i$| are pre-taut (Lemma 2.1b), then all |${\mathcal B}_i'$| are pre-taut (Lemma 2.1a), and then |${\mathcal B}'(q)$| is pre-taut (Lemma 2.1c). (b) If |${\mathcal B}$| is taut, then all |${\mathcal B}_i$| are taut (Lemma 2.1b), and then all |${\mathcal B}_i'$| are taut (Lemma 2.1a). (c) As |${\mathcal B}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$|⁠, we have also |${\mathcal M}_{\mathcal B}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$|⁠. (d) If |${\mathcal B}$| is Behrend, then at least one of the sets |${\mathcal C}_i$| is Behrend [10, Corollary 0.14], and so |${d({\mathcal M}_{{\mathcal C}_i})=1}$|⁠. Otherwise |$1\in{\mathcal B}$|⁠, so that |$1\in{\mathcal C}_i$| for some |$i$|⁠, whence |${\mathcal M}_{{\mathcal C}_i}={\mathbb Z}$|⁠. ■ Lemma 2.4. (compare [4, Proposition 4.25]) Assume that |${\mathcal B}\subseteq{\mathbb N}$| is taut and |$d({\mathcal M}_{\mathcal C})=1$| for some |${\mathcal C}\subseteq{\mathbb Z}$|⁠. If |$q\,{\mathcal C}\subseteq{\mathcal M}_{\mathcal B}$| for some |$q\geqslant1$|⁠, then |$b\mid q$| for some |$b\in{\mathcal B}$|⁠. □ Proof. By Lemma 2.2, |${\mathcal M}_{\mathcal C}\subseteq{\mathcal M}_{{\mathcal B}'(q)}$|⁠, so that |$d({\mathcal M}_{{\mathcal B}'(q)})=1$|⁠. Then |${\mathcal B}'(q)=\{1\}$| or |${\mathcal B}'(q)$| is not taut (Lemma 2.1d). If |${\mathcal B}'(q)=\{1\}$|⁠, then |$\gcd(b,q)=b$| for all |$b\in{\mathcal B}$|⁠, i.e |$b\mid q$| for all |$b\in{\mathcal B}$|⁠, which is impossible because |${\mathcal B}$| is infinite. Hence |${\mathcal B}'(q)$| is not taut. On the other hand, as |${\mathcal B}$| is taut by assumption, |${\mathcal B}'(q)$| is a finite union of taut sets |${\mathcal B}_i'$| (Lemma 2.3b). As |$d({\mathcal M}_{{\mathcal B}'(q)})=1$|⁠, also |$d({\mathcal M}_{{\mathcal B}_i'})=1$| for at least one of the sets |${\mathcal B}_i'$| (Lemma 2.3d), so that |${\mathcal B}_i'=\{1\}$| for this set (Lemma 2.1d). This implies |$q\in{\mathcal M}_{\mathcal B}$|⁠. ■ Recall that the topology on |$H$| is generated by the (open and closed) cylinder sets \begin{equation*} U_S(h):=\{h'\in H: \forall b\in S: h_b=h'_b\},\text{ defined for finite }S\subset{\mathcal B}\text{ and }h\in H, \end{equation*} and recall also the definition of |${\mathcal A}_S:=\{\gcd(b,{\mathrm{lcm}}(S)):b\in{\mathcal B}\}$|⁠. Note that |${\mathcal A}_S$| is finite and |$S\subseteq{\mathcal A}_{\mathcal S}$|⁠. Lemma 2.5. Let |$U=U_S(\Delta(n))$| for some |$S\subset{\mathcal B}$| and |$n\in{\mathbb Z}$|⁠. (a) If |$n\in{\mathcal M}_S$|⁠, then |$U\cap W=\emptyset$|⁠. (b) If |$U\cap W=\emptyset$|⁠, then |$n+{\mathrm{lcm}}(S)\cdot{\mathbb Z}\subseteq {\mathcal M}_{{\mathcal B}\cap{\mathcal A}_S}$|⁠. (c) There is a filtration of |${\mathcal B}$| by finite sets |$S$| for all of which |${\mathcal B}\cap{\mathcal A}_S= S$|⁠. (d) If |${\mathcal B}\cap{\mathcal A}_S= S$|⁠, then |$n\in{\mathcal M}_S$| iff |$U\cap W=\emptyset$| iff |$n+{\mathrm{lcm}}(S)\cdot{\mathbb Z}\subseteq{\mathcal M}_S$|⁠. □ Proof. (a) This follows immediately from the definitions of |$U_S(\Delta(n))$| and |$W$|⁠. (b) For each |$h\in U$| there is |$b\in{\mathcal B}$| such that |$h_b=0$|⁠. As |$U$| is compact, the Heine–Borel argument produces a finite set |$S'\subset{\mathcal B}$| such that for each |$h\in U$| there is |$b\in S'$| such that |$h_b=0$|⁠. Let |$s={\mathrm{lcm}}(S)$|⁠. This observation applies in particular to all |$h\in\Delta(n+s{\mathbb Z})\subseteq U_{S}(\Delta(n))=U$|⁠. That means, for each |$k\in{\mathbb Z}$| there is |$b_k\in S'$| such that |$b_k\mid n+sk$|⁠. In other words: |$n+s{\mathbb Z}\subset{\mathcal M}_{S'}$|⁠. The set |$S'$| need not be primitive automatically, but we can replace it w.l.o.g. by a primitive subset without changing its set of multiples. Then, as |$S'$| is finite, it is taut. Denote |$q=\gcd(n,s)$| and |${\mathcal C}=\frac{n}{q}+\frac{s}{q}\cdot{\mathbb Z}$|⁠. Then |$q\,{\mathcal C}=n+s{\mathbb Z}\subseteq{\mathcal M}_{S'}$|⁠, and as |$\gcd(n/q,s/q)=1$|⁠, |$d({\mathcal M}_{\mathcal C})=1$| (Dirichlet’s theorem, see [4, Corollary 4.24]). Now Lemma 2.4 shows that |$b\mid q=\gcd(n,s)$| for some |$b\in S'$|⁠. In particular, |$n\in b{\mathbb Z}$| and |$b\mid s={\mathrm{lcm}}(S)$| for that |$b\in S'$|⁠, so that |$b=\gcd(b,s)\in{\mathcal B}\cap{\mathcal A}_S$| and |$n+{\mathrm{lcm}}(S)\cdot{\mathbb Z}\subseteq b{\mathbb Z}\subseteq{\mathcal M}_{{\mathcal B}\cap{\mathcal A}_S}$|⁠. (c) It suffices to prove that for any finite |$S\subset{\mathcal B}$| there exists a finite |$S'\subset{\mathcal B}$| with |$S\subseteq S'$| and |${\mathcal B}\cap{\mathcal A}_{S'}=S'$|⁠. So let |$S\subset{\mathcal B}$| and |$S':={\mathcal B}\cap{\mathcal A}_S$|⁠. |$S'$| is finite, because |${\mathcal A}_S$| is finite, and obviously |$S\subseteq S'\subseteq{\mathcal B}\cap{\mathcal A}_{S'}$|⁠. As each |$b'\in S'\subseteq{\mathcal A}_S$| divides |${\mathrm{lcm}}(S)$|⁠, also |${\mathrm{lcm}}(S')$| divides |${\mathrm{lcm}}(S)$|⁠. Therefore |${\mathrm{lcm}}(S')={\mathrm{lcm}}(S)$|⁠, so that |${\mathcal A}_{S'}={\mathcal A}_S$|⁠. Hence |$S'={\mathcal B}\cap{\mathcal A}_{S'}$|⁠. (d) This follows from (a) and (b). ■ 2.2 Proof of Theorem A Let |${\mathcal B}=\{b_1,b_2,\dots\}$| be primitive, and denote |$S_1\subset S_2\subset\dots\subset{\mathcal B}$| a filtration of |${\mathcal B}$| by finite sets |$S_k$|⁠. Let |$s_k={\mathrm{lcm}}(S_k)$|⁠. We can assume without loss of generality that |$b\mid s_k\Rightarrow b\in S_k$| holds for all |$b\in{\mathcal B}$| and all |$k\in{\mathbb N}$|⁠. For each |$k\in{\mathbb N}$|⁠, the collection of all cylinder sets |$U_{S_k}(h)$|⁠, |$h\in H$|⁠, can be written explicitly as \begin{equation*} {\mathcal Z}_k:=\left\{U_{S_k}(\Delta(n)): n=1,\dots,s_k\right\}\!. \end{equation*} Suppose first that |${\mathcal B}$| is not taut. Then it contains a scaled copy |$c{\mathcal A}$| of a Behrend set |${\mathcal A}\subseteq\{2,3,\dots\}$|⁠. Enlarging |${\mathcal A}$|⁠, if necessary, we can assume that |$c{\mathcal A}={\mathcal B}\cap c{\mathbb Z}$|⁠. (As |${\mathcal B}$| is primitive, also the enlarged |${\mathcal A}$| does not contain the number |$1$|⁠). Let |$a_0>1$| be the smallest element of |${\mathcal A}$| and denote |$b_0=ca_0$|⁠. Let |$H_0=\{h\in H:h_{b_0}\in c{\mathbb Z}\}$|⁠. Then |$H_0$| is open and closed, and we will show that |$H_0\cap W\neq\emptyset$| but |$m_H(H_0\cap W)=0$|⁠, so that |$W$| is not Haar regular. First observe that |$(\Delta(c))_{b_0}=c\in c{\mathbb Z}$|⁠, so that |$\Delta(c)\in H_0$|⁠. Suppose for a contradiction that |$H_0\cap W=\emptyset$|⁠. Then |$\Delta(c)\not\in W$|⁠, that is there is |$b\in{\mathcal B}$| such that |$c\in b{\mathbb Z}$|⁠. Hence |$c{\mathcal A}\subseteq b{\mathbb Z}$|⁠, so that |$c{\mathcal A}=\{b\}$|⁠, because |$b\in{\mathcal B}$| and |${\mathcal B}$| is primitive. Hence |$b=ca_0=b_0$|⁠, so that |${\mathcal A}=\{a_0\}$|⁠, a contradiction, as |${\mathcal A}$| is Behrend. We turn to the proof of |$m_H(H_0\cap W)=0$|⁠. Let |${\mathcal H}_W^\ell=\{n\in\{0,\dots,s_\ell-1\}: U_{S_\ell}(\Delta(n))\cap H_0\cap W\neq\emptyset\}$|⁠. It suffices to show that |$\sum_{n\in{\mathcal H}_W^\ell}m_H(U_{S_\ell}(\Delta(n))\to0$| as |$\ell\to\infty$|⁠. As all cylinder sets |$U_{S_\ell}(\Delta(n))$| have identical Haar measure |$s_\ell^{-1}$|⁠, this is equivalent to |$\#{\mathcal H}_W^\ell/s_\ell\to0$| as |$\ell\to\infty$|⁠. So let |$\ell$| be so large that |$b_0\in S_\ell$|⁠. Denote |${\mathcal A}^\ell=\{a\in{\mathcal A}:ca\mid s_\ell\}$|⁠. As |$c{\mathcal A}\subseteq{\mathcal B}$|⁠, the sequence |$({\mathcal A}^\ell)_\ell$| is increasing and exhausts the set |${\mathcal A}$|⁠. If |$n\in{\mathcal H}_W^\ell$|⁠, then |$n\in c{\mathbb Z}$| and, by Lemma 2.5a, |$n\in{\mathcal F}_{S_\ell}$|⁠. Hence |$n=cn'\in{\mathcal F}_{S_\ell}$| for some |$n'\in{\mathbb Z}$|⁠. Suppose for a contradiction that |$n'\in{\mathcal M}_{{\mathcal A}^\ell}$|⁠, that is there are |$k\in{\mathbb Z}$| and |$a\in{\mathcal A}_\ell$| such that |$n'=ka$|⁠. Then |$n=kca$|⁠, where |$ca\in{\mathcal B}$| and |$ca\mid s_\ell$|⁠, so that |$ca\in S_\ell$|⁠, which contradicts |$n\in{\mathcal F}_{S_\ell}$|⁠. Hence |$n'\in{\mathcal F}_{{\mathcal A}^\ell}$| so that |$n\in c{\mathcal F}_{{\mathcal A}^\ell}=c\,({\mathbb Z}\setminus{\mathcal M}_{{\mathcal A}^\ell})$|⁠. As |${\mathcal A}$| is Behrend, |$\bar{d}({\mathbb Z}\setminus{\mathcal M}_{{\mathcal A}^\ell})\to0$| as |$\ell\to\infty$|⁠. Hence, \begin{equation*} \#{\mathcal H}_W^\ell/s_\ell \leqslant \#\left(c({\mathbb Z}\setminus{\mathcal M}_{{\mathcal A}^\ell})\cap[0,s_l)\right)/s_\ell \le \#\left(({\mathbb Z}\setminus{\mathcal M}_{{\mathcal A}^\ell})\cap[0,s_l)\right)/s_\ell = d({\mathbb Z}\setminus{\mathcal M}_{{\mathcal A}^\ell}) \to0. \end{equation*} Suppose now that |${\mathcal B}$| is taut. We must show that for any |$k\in{\mathbb N}$| and |$U\in{\mathcal Z}_k$| \begin{equation*} U\cap W=\emptyset\quad\text{or}\quad m_H(U\cap W)>0. \end{equation*} So fix some |$U=U_{S_k}(\Delta(n))$| such that |$m_H(U\cap W)=0$|⁠. We have to show that |$U\cap W=\emptyset$|⁠. Observe first that |$U_{S_k}(\Delta(m))=U$| if and only if |$m\in s_k{\mathbb Z}+n$|⁠. For |$\ell>k$| let \begin{equation*} {\mathcal G}_\ell:=(s_k{\mathbb Z}+n)\cap \left\{m\in{\mathbb Z}: U_{S_\ell}(\Delta(m))\cap W=\emptyset\right\} = (s_k{\mathbb Z}+n)\cap {\mathcal M}_{S_\ell}, \end{equation*} where we used Lemma 2.5c for the last equality. Observe that \begin{equation*} {\mathcal G}_\ell = {\mathcal G}_\ell+s_\ell{\mathbb Z} = \left({\mathcal G}_\ell\cap[0,s_\ell)\right)+s_\ell{\mathbb Z}. \end{equation*} Hence, for each |$\ell>k$|⁠, \begin{align*} \underline{d}\left((s_k{\mathbb Z}+n)\cap{\mathcal M}_{\mathcal B}\right) &= \liminf_{t\to\infty}\frac{\# \left( (s_k{\mathbb Z}+n)\cap {\mathcal M}_{\mathcal B}\cap[0,t) \right)}{t}\\ & \geqslant \liminf_{t\to\infty}\frac{\# \left((s_k{\mathbb Z}+n)\cap {\mathcal M}_{S_\ell}\cap[0,t) \right)}{t}\\ & = \liminf_{t\to\infty}\frac{\# \left({\mathcal G}_\ell\cap[0,t)\right)}{t} = \frac{\# \left({\mathcal G}_\ell\cap[0,s_\ell)\right)}{s_\ell}. \end{align*} As all |$U'\in{\mathcal Z}_\ell$| have identical Haar measure |$m_H(U')=s_\ell^{-1}$| and as |$m_H(U\setminus W)=m_H(U)$| by assumption, it follows that \begin{equation*} \begin{split} \underline{d}\left((s_k{\mathbb Z}+n)\cap{\mathcal M}_{\mathcal B}\right) &\geqslant \limsup_{\ell\to\infty}\frac{\# \left({\mathcal G}_\ell\cap[0,s_\ell)\right)}{s_\ell} = \limsup_{\ell\to\infty}m_H\left(\bigcup_{U'\in{\mathcal Z}_\ell,U'\subseteq U\setminus W}U'\right)\\ &= m_H(U\setminus W) =m_H(U) =s_k^{-1} = d(s_k{\mathbb Z}+n), \end{split} \end{equation*} so that, \begin{equation*} d\left((s_k{\mathbb Z}+n)\cap{\mathcal M}_{\mathcal B}\right)=d(s_k{\mathbb Z}+n). \end{equation*} Let |$q=\gcd(s_k,n)$|⁠, |$a'=s_k/q$| and |$r'=n/q$|⁠. Then |$\gcd(a',r')=1$| and |$q{\mathbb Z}\cap{\mathcal M}_{\mathcal B}=q{\mathbb Z}\cap{\mathcal M}_{q\cdot{\mathcal B}'(q)}$|⁠, in particular |$(s_k{\mathbb Z}+n)\cap{\mathcal M}_{\mathcal B}=(s_k{\mathbb Z}+n)\cap{\mathcal M}_{q\cdot{\mathcal B}'(q)}$|⁠. Hence, \begin{equation*} \begin{split} d\left((a'{\mathbb Z}+r')\cap{\mathcal M}_{{\mathcal B}'(q)}\right) &= q\cdot d\left(q\cdot\left((a'{\mathbb Z}+r')\cap{\mathcal M}_{{\mathcal B}'(q)}\right)\right) = q\cdot d\left((s_k{\mathbb Z}+n)\cap {\mathcal M}_{q\cdot{\mathcal B}'(q)}\right)\\ &= q\cdot d\left((s_k{\mathbb Z}+n)\cap {\mathcal M}_{{\mathcal B}}\right) = q\cdot d(s_k{\mathbb Z}+n) = q\cdot d\left(q(a'{\mathbb Z}+r')\right)\\ &= d(a'{\mathbb Z}+r') =1/a'. \end{split} \end{equation*} In view of Lemma 1.17 in [10], this suffices to conclude that |${\mathcal B}'(q)$| is Behrend. On the other hand, as |${\mathcal B}$| is taut, |${\mathcal B}'(q)$| is pre-taut (Lemma 2.3), so that |$1\in{\mathcal B}'(q)$| or |${\mathcal B}'(q)$| is not Behrend (Lemma 2.1e). Hence |$1\in{\mathcal B}'(q)$|⁠. This implies |$q\in{\mathcal M}_{\mathcal B}$|⁠, which in turn implies |$U\cap W=U_{S_k}(\Delta(n))\cap W=\emptyset$| (the property to be proved): Indeed, if |$q\in{\mathcal M}_{\mathcal B}$|⁠, then there is some |$b\in{\mathcal B}$| with |$b\mid q$|⁠, and as |$q\mid s_k$|⁠, this implies |$b\mid s_k$|⁠, so that |$b\in S_k$|⁠. From |$b\mid q\mid n$| we then conclude that |$n\in{\mathcal M}_{S_k}$|⁠, and Lemma 2.5a implies |$U_{S_k}(\Delta(n))\cap W=\emptyset$|⁠. It remains to show that the implication |$(i)$||$\Rightarrow$||$(iii)$| follows from Proposition 1.1, which will be proved in the next subsection. So let |$h\in W$|⁠. By the proposition there exists |$n\in{\mathcal F}_{\mathcal B}$| such that |$\Delta(n)\in U_S(h)$|⁠, hence |$\Delta(n)\in U_S(h)\cap(\Delta({\mathbb Z})\cap W)$|⁠. As this holds for all finite |$S\subset{\mathcal B}$|⁠, this proves the claim. 2.3 Tautification of the set |${\mathcal B}$| and regularization of the window |$W$| In [4, Section 4.2] the authors provide a construction that associates to each (non-taut) set |${\mathcal B}$| a taut set |${\mathcal B}'$| such that |${\mathcal F}_{{\mathcal B}'}\subseteq{\mathcal F}_{\mathcal B}$| but |$\overline{d}({\mathcal F}_{\mathcal B}\setminus{\mathcal F}_{{\mathcal B}'})=0$|⁠, and such that the two Mirsky measures |$\nu_\eta$| and |$\nu_{\eta'}$| determined by |${\mathcal B}$| and |${\mathcal B}'$| coincide. |${\mathcal B}$| and |${\mathcal B}'$| determine groups |$H$| resp. |$H'$| with windows |$W$| resp. |$W'$|⁠, and while the window |$W$| is not Haar regular (if |${\mathcal B}$| is non-taut), the window |$W'$| is Haar regular because of Theorem A. On the abstract level one can also pass from the window |$W\subseteq H$| to its Haar regularization|$W_{reg}:={\mathrm{supp}}(m_H|_W)$| (introduced in [14]), which also determines the same Mirsky measure on |$\{0,1\}^{\mathbb Z}$|⁠. However, |$W_{reg}$| will not be a window of the particular arithmetic type defined in (2), in particular it need not be aperiodic. The construction of |${\mathcal B}'$| given |${\mathcal B}$| in [4] suggests an obvious factor map |$f:H\to H'$|⁠, and we expect that also |$f(W_{reg})=W'$|⁠, so that in this sense the regularization of |$W$| and the tautification of |${\mathcal B}$| are two sides of the same medal. The following example illustrates this discussion. Example 2.1. Let |${\mathcal P}=\{p_1,p_2,\ldots\}$| denote the set of primes. Let |${\mathcal B}:=\bigcup_{i\geq1}p_i^2({\mathcal P}\setminus\{p_i\})$|⁠. Note that |${\mathcal B}$| is primitive. It is not taut, because it contains rescalings of Behrend sets. The corresponding taut set is |${\mathcal B}'=\{p_i^2:\:i\geq1\}$|⁠, which generates the square-free system. (Note that |$\eta(n)=1$| at all square-free numbers and also at |$p_i^k$| for |$i\geq1$| and |$k\geq2$|⁠.) □ 2.4 The property |$\overline{\Delta({\mathbb Z})\cap W}=W$| Proof of Proposition 1.1. Given |$h\in W$|⁠, we need to show that for each finite |$S\subset{\mathcal B}$| the set \begin{equation*} {\mathcal L}_S(h) := \left\{n\in{\mathcal F}_{\mathcal B}:\text{ $h_b=n$ mod $b$ for each $b\in S$}\right\} \end{equation*} has asymptotic density |$m_H(U_S(h)\cap W)>0$|⁠. By Theorem A, the tautness assumption on |${\mathcal B}$| implies that |$W$| is Haar regular, so that indeed \begin{equation*} m_H(U_S(h)\cap W)>0. \end{equation*} Let |${\mathcal B}=\{b_1,b_2,\ldots\}$| and, for |$K\geq 1$|⁠, |$W_K:=\{g\in H:\:g_i\neq0\text{ for }i=1,\ldots,K\}$|⁠. Then |$W_K$| is clopen and |$W\subseteq W_K$|⁠. Moreover, |$W_{K}\supseteq W_{K+1}$| and |$\bigcap_KW_K=W$|⁠. Fix |$\varepsilon>0$|⁠. We now choose |$K\geq1$| so that \begin{equation}\label{tcrt1} m_H(W_K\setminus W)<\varepsilon. \end{equation} (11) Denote by |$R$| the rotation |$R_{\Delta(1)}$| on |$H$|⁠. Since |$U_S(h)\cap W_K$| is clopen (and |$R$| is strictly ergodic), \begin{align}\label{tcrt2} \left|\frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{U_S(h)\cap W_K}(R^n\Delta(0))-m_H\left(U_S(h)\cap W_K\right)\right|<\varepsilon \end{align} (12) for all |$N\geq N_0$|⁠. Moreover, we can choose |$N_1$| so that for |$N\geq N_1$|⁠, we also have \begin{equation}\label{tcrt3} \left|\frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{U_S(h)\cap W_K}(R^n\Delta(0))-\frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{U_S(h)\cap W}(R^n\Delta(0))\right|<\varepsilon. \end{equation} (13) Indeed, if \begin{equation*} R^n\Delta(0)=\Delta(n)\in \left(U_S(h)\cap W_K\right)\setminus \left(U_S(h)\cap W\right)\subset W_K\setminus W, \end{equation*} then (by setting |${\mathcal B}_K=\{b_1,\ldots,b_K\}$|⁠), we have \begin{equation*} n\in {\mathcal F}_{{\mathcal B}_K}\cap {\mathcal M}_{{\mathcal B}}={\mathcal M}_{{\mathcal B}}\setminus{\mathcal M}_{{\mathcal B}_K}. \end{equation*} Therefore, by the Davenport–Erdös theorem [10, Eq. (0.67)], we can choose first |$K\geq1$| sufficiently large so that |$\overline{d}({\mathcal M}_{{\mathcal B}}\setminus{\mathcal M}_{{\mathcal B}_K})<\varepsilon$| and then |$N_1$| so that \begin{equation*} \frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{W_K\setminus W}(R^n\Delta(0)) = \frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{{\mathcal M}_{{\mathcal B}}\setminus{\mathcal M}_{{\mathcal B}_K}}(n)<\varepsilon \end{equation*} for all |$N\geq N_1$|⁠, so in particular (13) holds. In view of (11), (12) and (13), it follows that \begin{equation*} \lim_{N\to\infty}\frac1N\sum_{n\leq N}{\rm 1}\kern-0.24em{\rm I}_{U_S(h)\cap W}(R^n\Delta(0))=m_H(U_S(h)\cap W). \end{equation*} As |$R^n\Delta(0)=\Delta(n)\in U_S(h)\cap W$| if and only if |$n\in{\mathcal L}_S(h)$|⁠, this finishes the proof. ■ Example 2.2. (⁠|$\overline{\Delta({\mathbb Z})\cap W}=W$| does not imply tautness) Suppose that |$(m_k,r_k)$|⁠, |${k\in{\mathbb N}}$|⁠, is an enumeration of all coprime pairs of natural numbers. For any |$k$| choose a prime |$p_k\in r_k+m_k{\mathbb Z}$| such that |$p_k>2^{k+1}$|⁠. Let |${\mathcal B}={\mathcal P}\setminus\{p_k:k\in{\mathbb N}\}$|⁠. Clearly |${\mathcal B}$| is primitive, and |${\mathcal M}_{\{p_k:k\in{\mathbb N}\} }$| has upper density less than or equal to |$\sum_{k=1}^{\infty}1/2^{k+1}=1/2$|⁠. Thus |$d({\mathcal M}_{{\mathcal B}})=1$| and |${\mathcal B}$| is not taut [10, Corollary 0.14]. But |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠. Indeed, let |$h=(h_b)_{b\in{\mathcal B}}\in W$| and take any finite set |$S\subset {\mathcal B}$|⁠. We are going to show that |$U_S(h)\cap W\cap \Delta({\mathbb Z})\neq \emptyset$|⁠. Let |$n\in{\mathbb Z}$| be such that |$n= h_b$| mod |$b$| for |$b\in S$|⁠. Since |$h\in W$|⁠, |$b$| does not divide |$n$| for any |$b\in S$|⁠, that is |${\mathrm{lcm}}(S)$| and |$n$| are coprime. Then |$({\mathrm{lcm}}(S),n)=(m_k,r_k)$| for some |$k$|⁠, and the prime number |$p_k$| belongs to arithmetic progression |$r_k+m_k{\mathbb Z}=n+{\mathrm{lcm}}(S){\mathbb Z}$|⁠, in other words |$\Delta(p_k)\in U_S(\Delta(n))=U_S(h)$|⁠. Finally, |$\Delta(p_k)\in W$|⁠, because the prime number |$p_k$| does not belong to |${\mathcal B}$| and hence also not to |${\mathcal M}_{\mathcal B}$|⁠. □ 2.5 |$X_\eta$| and |$X_\varphi$| The set |${\mathcal B}\subseteq{\mathbb N}$| has light tails, if \begin{equation} \lim_{K\to\infty}\overline{d}\left({\mathcal M}_{\{b\in{\mathcal B}:b>K\}}\right)=0. \end{equation} (14) If |${\mathcal B}$| has light tails, then |${\mathcal B}$| is taut, but the converse does not hold [4, Section 4.3]. Here we prove: Proposition 2.2. If |${\mathcal B}$| has light tails, then |$X_\eta= X_\varphi$|⁠. □ Proof. Let |$H=(h_k)\in H$| and |$n\in{\mathbb N}$|⁠. We are going to show that |$\varphi(h)[-n,n]=\eta[l+1,l+2n+1]$| for some |$l\in{\mathbb Z}$|⁠. We know that |$\varphi(h)(i)=1$| if and only if |$h_j+i$| is not a multiple of |$b_j$| for any |$j\in{\mathbb N}$|⁠. For any |$i\in[-n,n]$| such that |$\varphi(h)(i)=0$| let |$k_i$| be such that |$b_{k_i}|h_{k_i}+i$|⁠. Let |$K\in{\mathbb N}$| be such that the set |$\mathscr{A}:=\{b_1,\ldots,b_K\}$| contains |$b_{k_i}$|⁠, for |$i\in[-n,n]$| and any |$b_k$| with |$k>K$| has a prime factor |$p>2n+1$|⁠. Since |$h\in H$|⁠, there exists |$m\in Z$| such that \begin{equation}\label{d1} m=h_k \mod b_k \end{equation} (15) for all |$k\le K$|⁠. It follows that \begin{equation*} ({\mathrm{supp}} \varphi(h)\cap [-n,n])+m=[-n+m,n+m]\cap {\mathcal F}_{{\mathcal A}}. \end{equation*} Indeed, if |$i\in {\mathrm{supp}} \varphi(h)\cap [-n,n]$|⁠, then |$h_k+i$| is not a multiple of |$b_k$| for any |$k\in {\mathbb N}$|⁠. By (15) we get that |$m+i$| is not a multiple of |$b_k$| for any |$k\le K$|⁠, that is, |$m+i\in {\mathcal F}_{{\mathcal A}}$|⁠. On the other hand, if |$i\notin {\rm supp} \varphi(h)\cap [-n,n]$|⁠, then |$b_{k_i}|h_{k_i}+i$|⁠. Since |$k_i\le K$|⁠, again by (15), we obtain |$b_{k_i}|m+i$|⁠, that is |$m+i\notin {\mathcal F}_{{\mathcal A}}$|⁠. We recall proposition 5.11 from [4]: Assume that |${\mathcal B}\subset {\mathbb N}$| has light tails and |${\mathcal B}^{(n)}\subset {\mathcal A}\subset {\mathcal B}$|⁠. Suppose that \begin{align}\label{dziedz:as1} \{k+1,\ldots,k+n\}\cap \mathcal{M}_{{\mathcal A}}=\{k+i_0,k+i_1,\ldots,k+i_r\} \end{align} (16) for some |$1\le i_0,\ldots,i_r\le n$|⁠, |$r<n$|⁠. Then the density of |$k'\in{\mathbb N}$| such that $$ \{k'+1,\ldots,k'+n\}\cap \mathcal{M}_{{\mathcal B}}=\{k'+i_0,k'+i_1,\ldots,k'+i_r\} $$ is positive. (Here |${\mathcal B}^{(n)}:=\{b\in {\mathcal B} : p\le n \text{ for any }p\in{\rm Spec}(b)\} $|⁠. If |${\mathcal B}$| is primitive, then |${\mathcal B}^{(n)}$| is finite.) Hence there exists |$l\in{\mathbb Z}$| such that \begin{equation*} ([-n+m,n+m]\cap {\mathcal F}_{{\mathcal A}})+l+n+1-m=[l+1,l+2n+1]\cap {\mathcal F}_{{\mathcal B}}. \end{equation*} It follows that |$\varphi(h)[-n,n]=\eta[l+1,l+2n+1]$|⁠. ■ We now present a Behrend set (hence a non-taut set), for which |$X_\eta$| is a strict subset of |$X_\varphi$|⁠. Example 2.3. Let |${\mathcal B}=\{p_2,p_3,\ldots\}=\{3,5,7,11,\ldots\}$| - the set of all odd prime numbers. Since we are in the coprime case, $$ H=\prod_{k=2}^\infty{\mathbb Z}/p_k{\mathbb Z}.$$ Now, |$\eta=\varphi(\Delta(0))$| is the characteristic function of the |${{\mathcal B}}$|-free set |$\{\pm 2^m:\:m\geq0\}$|⁠. We compute an initial block of |$\varphi(h)$| for $$ h=(0,1,0,0,\ldots)\in H.$$ We have |$\varphi(h)(0)=0$|⁠, |$\varphi(h)(1)=1$|⁠, |$\varphi(h)(2)=1$|⁠, |$\varphi(h)(3)=0$|⁠, |$\varphi(h)(4)=0 $| (If we add 4 to each coordinate of |$h$|⁠, we obtain the sequence |$(1,0,4,4,\ldots)$|⁠, whence |$\varphi(h)(4)=0$|⁠.), |$\varphi(h)(5)=1$|⁠, |$\varphi(h)(6)=0$|⁠, |$\varphi(h)(7)=0$| and |$\varphi(h)(8)=1$|⁠. It follows that the block |$11001001$| appears on |$\varphi(h)$|⁠. But there is no block |$\underline{a}$| of length 8 appearing on |$\eta$| and such that |$11001001\le\underline{a}$|⁠. Indeed, the two neighbouring 1′s at the beginning of |$\underline{a}$| could only appear at the positions 1,2 or -2,-1 in |$\eta$|⁠. In the both cases this would force |$\eta(5)=1$|⁠, which is not true. This shows that |$\varphi(h)\not\in X_\eta$|⁠, although it belongs to |$X_\varphi$|⁠. (Indeed, |$\varphi(h)$| does not even belong to |$\widetilde X_\eta$|⁠, the hereditary closure of |$X_\eta$|⁠, see [4].) □ Question 2.1. If |${\mathcal B}$| is taut, is then |$X_\eta=X_\varphi$|? (We recall that in case of |${\mathcal B}$| taut, the Mirsky measure is supported on |$X_\eta$|⁠.) □ 3 Minimality/Proximality of |$\boldsymbol{X_\eta}$| and Topological Properties of |$\boldsymbol{W}$| Throughout this section we assume that |${\mathcal B}$| is primitive. 3.1 Arithmetic of |${\mathcal B}$| and topology of |$W$|⁠, part II Recall from (5) that |${\mathcal A}_S:=\{\gcd(b,{\mathrm{lcm}}(S)): b\in{\mathcal B}\}$| and |${\mathcal F}_{{\mathcal A}_S}\subseteq{\mathcal F}_{\mathcal B}$| for |$S\subset{\mathcal B}$|⁠. If |$S\subseteq S'\subset{\mathcal B}$|⁠, then the following inclusions and implications are obvious: \begin{equation}\label{eq:inclusions} S\subseteq S'\subseteq{\mathcal A}_{S'}\subseteq{\mathcal M}_{{\mathcal A}_S} \Rightarrow{\mathcal M}_S\subseteq{\mathcal M}_{S'}\subseteq {\mathcal M}_{{\mathcal A}_{S'}}\subseteq{\mathcal M}_{{\mathcal A}_S} \Rightarrow {\mathcal F}_{{\mathcal A}_S}\subseteq{\mathcal F}_{{\mathcal A}_{S'}}\subseteq{\mathcal F}_{S'}\subseteq{\mathcal F}_S. \end{equation} (17) Let |${\mathcal E}:=\bigcup_{S\subset{\mathcal B}}{\mathcal F}_{{\mathcal A}_S}$| and observe that |${\mathcal E}\subseteq{\mathcal F}_{\mathcal B}$|⁠. Lemma 3.1. (a) For all |$S\subset{\mathcal B}$| and |$n\in{\mathbb Z}$| we have: |$U_S(\Delta(n))\subseteq W\Leftrightarrow n\in{\mathcal F}_{{\mathcal A}_S}$|⁠. (b) If |$(S_k)_k$| is a filtration of |${\mathcal B}$| with finite sets and |$\lim_k\Delta(n_{S_k})=h$| (see Remark 5.2), then |$h\in {\mathrm{int}}(W)$| if and only if |$n_{S_k}\in{\mathcal F}_{{\mathcal A}_{S_k}}$| for some |$k$|⁠. (c) For all |$n\in{\mathbb Z}$| we have: |$\Delta(n)\in{{\mathrm{int}}(W)}\Leftrightarrow n\in{\mathcal E}$|⁠. (d) |${\mathrm{int}}(W)=\emptyset\Leftrightarrow {\mathcal E}=\emptyset\Leftrightarrow \forall S\subset{\mathcal B}: {\mathcal F}_{{\mathcal A}_S}=\emptyset \Leftrightarrow \forall S\subset{\mathcal B}: 1\in{\mathcal A}_S.$| □ Proof. (a) As |$U_S(\Delta(n))$| is clopen, \begin{equation*} \begin{split} U_S(\Delta(n))\not\subseteq W &\Leftrightarrow \exists m\in n+{\mathrm{lcm}}(S)\cdot{\mathbb Z}\ \exists c\in{\mathcal B}: c\mid m\\ &\Leftrightarrow \exists c\in{\mathcal B}\ \exists k\in{\mathbb Z}: c\mid n+k\cdot{\mathrm{lcm}}(S)\\ &\Rightarrow \exists c\in{\mathcal B}:\ \gcd(c,{\mathrm{lcm}}(S))\mid n\\ &\Leftrightarrow \exists k\in{\mathcal A}_S: k\mid n\\ &\Leftrightarrow n\not\in{\mathcal F}_{{\mathcal A}_S}. \end{split} \end{equation*} That the only implication is also an equivalence is a consequence of the CRT. Indeed, if |$\gcd(c,{\mathrm{lcm}}(S))\mid n$|⁠, then there exist |$k,l\in{\mathbb Z}$| such that |$l\cdot c-k\cdot{\mathrm{lcm}}(S)=n$|⁠, thus |$c\mid n+k\cdot{\mathrm{lcm}}(S)$|⁠. (b) Assume that |$h\in {\mathrm{int}}(W)$|⁠, that is |$U_S(h)\subseteq W$| for some |$S$|⁠. Then, for |$k$| such that |$S\subseteq S_k$|⁠, we have |$U_{S_k}(\Delta(n_{S_k}))=U_{S_k}(h)\subseteq W$|⁠, which is equivalent to |$n_{S_k}\in{\mathcal F}_{{\mathcal A}_{S_k}}$| by a). Conversely, if |$n_{S_k}\in{\mathcal F}_{{\mathcal A}_{S_k}}$| then, again by a), |$U_{S_k}(\Delta(n_{S_k}))=U_{S_k}(h)\subseteq W$| and |$h\in{\mathrm{int}}(W)$|⁠. (c) Follows from (a). (d) Follows from (c). ■ Recall from (6) that |${{\mathcal A}_\infty}:=\{n\in{\mathbb N}: \forall_{S\subset{\mathcal B}}\ \exists_{S': S\subseteq S'}: n\in{\mathcal A}_{S'}\setminus S'\}$|⁠. Lemma 3.2. (a) If |$(S_k)_k$| is a filtration of |${\mathcal B}$| with finite sets, then \begin{equation*} \limsup_{k\to\infty}{\mathcal A}_{S_k}\setminus S_k={{\mathcal A}_\infty}. \end{equation*} (b) For each |$n\in{{\mathcal A}_\infty}$| there is a filtration |$(S_k)_k$| of |${\mathcal B}$| with finite sets such that \[ n\in\bigcap_{k\in{\mathbb N}}{\mathcal A}_{S_k}\setminus S_k. \] □ Proof. (a) Assume that |$n\in{\mathcal A}_{S_k}\setminus S_k$| for infinitely many |$k$|⁠, and let |$S\subset{\mathcal B}$|⁠. Then there is |$k$| such that |$S\subseteq S_k$| and |$n\in{\mathcal A}_{S_k}\setminus S_k$|⁠. Hence |$n\in{{\mathcal A}_\infty}$|⁠. Conversely, let |$n\in{\mathcal A}_{\infty}$|⁠. There is a finite set |$S_1$| such that |$n\in{\mathcal A}_{S_1}\setminus S_1$|⁠. Assume that we have constructed sets |$S_1\subset S_2\subset \ldots \subset S_k$| with the property that |$n\in{\mathcal A}_{S_i}\setminus S_i$| for |$i=1,\ldots,k$| and |$\{1,\ldots,k\}\cap{\mathcal B}\subset S_k$|⁠. Then there is a set |$S_{k+1}$| containing |$S_{k}\cup(\{k+1\}\cap {\mathcal B})$| and such that |$n\in{\mathcal A}_{S_{k+1}}\setminus S_{k+1}$|⁠. In this inductive way we construct a filtration |$(S_k)_k$| as required. (b) follows from (a). ■ Lemma 3.3. The sets |${\mathcal E}$| and |${{\mathcal A}_\infty}$| are related by the identity \[ {\mathcal E}={\mathcal F}_{{\mathcal B}\cup{{{\mathcal A}_\infty}}}={\mathcal F}_{\mathcal B}\cap{\mathcal F}_{{{\mathcal A}_\infty}}. \] □ Proof. Let |$n\in{\mathcal E}$| and chose |$S$| such that |$n\in{\mathcal F}_{{\mathcal A}_S}$|⁠. Take arbitrary |$b\in{\mathcal B}$| and |$c\in {{\mathcal A}_\infty}$|⁠. There exists a finite set |$S'$| such that |$S\cup\{b\}\subseteq S'$| and |$c\in{\mathcal A}_{S'}\setminus S'$|⁠. Since |${\mathcal F}_{{\mathcal A}_S}\subseteq {\mathcal F}_{{\mathcal A}_{S'}}$|⁠, |$n\in {\mathcal F}_{{\mathcal A}_{S'}}$|⁠, hence neither |$b$| nor |$c$| divides |$n$|⁠. We have proved that |${\mathcal E}\subseteq {\mathcal F}_{{\mathcal B}\cup{{\mathcal A}_\infty}}$|⁠. In order to prove the other inclusion assume that |$n\in {\mathbb N}$| and that for any |$S$| there exists |$c_S\in{\mathcal A}_S$| dividing |$n$|⁠. As |$n$| has only finitely many divisors, it has a divisor |$c$| such that there exists a filtration |$(S_k)_k$| of |${\mathcal B}$| such that |$c\in{\mathcal A}_{S_k}$| for any |$k\in {\mathbb N}$|⁠. If |$c\notin{\mathcal B}$|⁠, then |$c\in{\mathcal A}_{S_k}\setminus S_k$| for any |$k\in {\mathbb N}$|⁠. This proves |$n\notin{\mathcal F}_{{\mathcal B}\cup{{\mathcal A}_\infty}}$|⁠. ■ Lemma 3.4. |${{\mathcal A}_\infty}=\emptyset$| if and only if |${\mathcal E}={\mathcal F}_{\mathcal B}$|⁠. □ Proof. If |${{\mathcal A}_\infty}=\emptyset$|⁠, then |${\mathcal E}={\mathcal F}_{\mathcal B}$| by Lemma 3.3. Conversely, assume that |${\mathcal E}={\mathcal F}_{\mathcal B}$|⁠. Then |${\mathcal F}_{\mathcal B}\subseteq{\mathcal F}_{{{\mathcal A}_\infty}}$| by Lemma 3.3, so that |${{\mathcal A}_\infty}\subseteq{\mathcal M}_{{{\mathcal A}_\infty}}\subseteq{\mathcal M}_{\mathcal B}$|⁠. Suppose for a contradiction that there exists some |$n\in{{\mathcal A}_\infty}$|⁠. Then there is |$b\in{\mathcal B}$| such that |$n\in b{\mathbb Z}$|⁠, that is |$b\mid n$|⁠, and there is a finite set |$S=S_k\subset{\mathcal B}$| such that |$n\in{\mathcal A}_{S}\setminus S$|⁠, see Lemma 3.2b. Hence there exists |$b'\in{\mathcal B}$| such that |$n=\gcd({\mathrm{lcm}}(S),b')$|⁠. It follows that |$b\mid n\mid b'$|⁠, which is impossible, because |${\mathcal B}$| is assumed to be primitive. ■ Proposition 3.1. The following conditions are equivalent: (i) |$W\neq\overline{{\mathrm{int}}(W)}$|⁠. (ii) For any filtration |$S_0\subset S_1\subset\ldots \subset {\mathcal B}$| of |${\mathcal B}$| with finite subsets |$S_k$|⁠, there exists a number |$d$| such that |$d\in {\mathcal A}_{S_k}\setminus S_k$|⁠, for infinitely many |$k\in{\mathbb N}$|⁠. (iii) There exists a filtration |$S_0\subset S_1\subset\ldots \subset {\mathcal B}$| of |${\mathcal B}$| with finite subsets |$S_k$| and there exists a number |$d$| such that |$d\in {\mathcal A}_{S_k}\setminus S_k$|⁠, for every |$k\in{\mathbb N}$|⁠. (iv) There are |$d\in{\mathbb N}$| and an infinite pairwise coprime set |${\mathcal A}\subseteq{\mathbb N}\setminus\{1\}$| such that |$d\,{\mathcal A}\subseteq{\mathcal B}$|⁠. □ Proof. |$(i)\Rightarrow (ii)$|⁠: Let |$h=(h_b)\in W\setminus \overline{{\mathrm{int}}(W)}$|⁠. There exists |$S$| such that |$U_S(h)\cap {\mathrm{int}}(W)=\emptyset$|⁠. We can assume that any |$b\in {\mathcal B}$| such that |$b|{\mathrm{lcm}}(S)$|⁠, belongs to |$S$|⁠. (Otherwise we can incorporate all such |$b$|’s into |$S$|⁠, there are finitely many of them.) Let |$n$| be a number such that \begin{equation}\label{f1} n= h_b \mod b \end{equation} (18) for |$b\in S$|⁠. Then |$\Delta(n+k{\mathrm{lcm}}(S))\in U_S(h)$|⁠, hence |$\Delta(n+k{\mathrm{lcm}}(S))\notin {\mathrm{int}}(W)$| for any |$k\in{\mathbb Z}$|⁠. This means (see Lemma 3.1) that for any finite set |$T$|⁠, in particular for any |$T=S_k$|⁠, the arithmetic progression |$n+{\mathrm{lcm}}(S){\mathbb Z}$| is contained in |${\mathcal M}_{{\mathcal A}_T}$|⁠. Since the set |${\mathcal A}_T$| is finite, it follows that |${\mathcal A}_T$| contains a divisor of |$\gcd(n,{\mathrm{lcm}}(S))$|⁠. (Apply Dirichlet’s theorem on primes in arithmetic progressions.) There is only finitely many divisors of |$\gcd(n,{\mathrm{lcm}}(S))$|⁠, hence one of them, denote it by |$d$|⁠, appears in |${\mathcal A}_{S_k}$| for infinitely many |$k$|⁠. To finish the proof it is enough to observe that |$d\notin{\mathcal B}$| (consequently, |$d\notin S_k$|⁠, for any |$k$|⁠). Indeed, otherwise |$d\in S$|⁠, by our assumption on |$S$|⁠. Moreover, |$d|n$| and then, by (18), |$d|h_b$|⁠, where |$b=d$|⁠, which leads to a contradiction with the assumption |$h\in W$|⁠. |$(ii)\Rightarrow (iii)$|⁠: obvious |$(iii)\Rightarrow (i)$|⁠: Assume that |$d\in{\mathcal A}_{S_k}\setminus S_k$| for any |$k$|⁠. Then |$d\notin {\mathcal M}_{{\mathcal B}}$|⁠. (Otherwise |$d$| is divisible by some |$b\in{\mathcal B}$|⁠. On the other hand, |$d$| divides some |$b'\in{\mathcal B}$| as a member of |${\mathcal A}_{S_k}$|⁠, which in view of the fact that |${\mathcal B}$| is primitive, leads to the conclusion that |$d=b=b'\in{\mathcal B}$|⁠. But it is not true, since |$d\notin S_k$| for any |$k$|⁠.) Hence |$\Delta(d)\in W$|⁠. We prove that |$\Delta(d)\notin\overline{{\mathrm{int}}(W)}$|⁠. It is enough to show that |$U_{S_0}(\Delta(d))\cap {\mathrm{int}}(W)=\emptyset$|⁠. Assume that |$h=(h_b)\in U_{S_0}(\Delta(d))\cap {\mathrm{int}}(W)$|⁠. It means that \begin{equation}\label{f2} d=h_b \mod b\; \mbox{for any}\;b\in S_0, \end{equation} (19) and there exists a finite set |$T\subset{\mathcal B}$| such that |$U_T(h)\subset W$|⁠. We can assume that |$T=S_k$| for some |$k$|⁠. Let |$m\in{\mathbb Z}$| be such that \begin{equation}\label{f3} m=h_b \mod b\; \mbox{for any}\;b\in S_k. \end{equation} (20) Let |$c\in {\mathcal B}$| be such that |$\gcd(c,{\mathrm{lcm}}(S_k))=d$|⁠. {Clearly, |$c\notin S_k$|⁠, since |$d\notin {\mathcal B}$|⁠.} Since |$U_{S_k}(h)\subset W$|⁠, it follows that there exists |$b\in S_k$| such that \begin{equation}\label{f4} \gcd(c,b)\; \mbox{does not divide}\; h_b. \end{equation} (21) Indeed, otherwise there would exist |$l\in{\mathbb Z}$| such that |$l\equiv h_b$| mod |$b$| for |$b\in S_k$| and |$l= 0$| mod |$c$|⁠, hence |$\Delta(l)\in U_{S_k}(h)$|⁠, but |$\Delta(l)\notin W$|⁠, a contradiction. Thus, in view of (20), \begin{equation}\label{f4a} \gcd(c,b) \; \mbox{does not divide}\; m. \end{equation} (22) On the other hand, \begin{equation}\label{f5} \gcd(c,b)|\gcd(c,{\mathrm{lcm}}(S_k))=d. \end{equation} (23) Since |$d\in{\mathcal A}_{S_0}\setminus S_0$|⁠, we get \begin{equation}\label{f6} d|{\mathrm{lcm}}(S_0). \end{equation} (24) By (19) and (20), \begin{equation}\label{f7} {\mathrm{lcm}}(S_0)|m-d. \end{equation} (25) Now, (23), (24) and (25) imply |$\gcd(c,b)|m$|⁠, a contradiction with (22). |$(iii)\Rightarrow (iv)$|⁠: Assume that |$d\in{\mathcal A}_{S_k}\setminus S_k$| for any |$k$|⁠. Then \begin{equation}\label{eq:search-for-coprime} \forall k\in{\mathbb N}\ \exists b_k\in{\mathcal B}\setminus S_k:\ d=\gcd(b_k,{\mathrm{lcm}}(S_k)). \end{equation} (26) As |$d\not\in S_k$|⁠, we have |$b_k\neq d$| for all |$k$|⁠. We choose a subsequence |$b_{k_1},b_{k_2},\dots$| of |$(b_k)_k$| in the following way: Let |$k_1=1$|⁠, and given |$k_1,\dots,k_j$|⁠, let \begin{equation*} k_{j+1}:=\min\left\{k\in{\mathbb N}: b_{k_1},\dots,b_{k_j}\in {S_{k}}\right\}\!. \end{equation*} Let |$a_j=b_{k_j}/d$| for all |$j\in{\mathbb N}$| and denote |${\mathcal A}=\{a_j:j\in{\mathbb N}\}$|⁠. Then |${\mathcal A}\subseteq{\mathbb N}$| and |$d\,{\mathcal A}\subseteq {\mathcal B}$| by construction. Suppose that |$1\in{\mathcal A}$|⁠. Then |$d\in{\mathcal B}$|⁠, a contradiction to (26), as |${\mathcal B}$| is primitive. Hence |${\mathcal A}\subseteq{\mathbb N}\setminus\{1\}$|⁠. It remains to prove that |${\mathcal A}$| is pairwise coprime. Suppose for a contradiction that there is a prime number |$p$| dividing some |$a_i$| and |$a_j$|⁠, |$i<j$|⁠. Then |$pd\mid b_{k_i}$| and |$pd\mid b_{k_j}$|⁠. As |$b_{k_i}\in S_{k_j}$|⁠, it follows that |$pd\mid{\mathrm{lcm}}(S_{k_j})$|⁠, so that |$pd\mid\gcd(b_{k_j},{\mathrm{lcm}}(S_{k_j}))=d$| (see (26)), which is impossible. |$(iv)\Rightarrow (iii)$|⁠: Let |$d\in{\mathbb N}$| and |${\mathcal A}=\{a_1<a_2<\dots\}$| be as in |$(iv)$|⁠. Then |$d\not\in{\mathcal B}$|⁠, because |${\mathcal B}$| is primitive. For |$k\in{\mathbb N}$| let |$S_k={\mathcal B}\cap\{1,\dots,k\}\cup\{da_k\}$|⁠. As all |$a_j$| are pairwise coprime, there are |$j_1<j_2<\dots\in{\mathbb N}$| such that |$a_{j_k}$| is coprime to |${\mathrm{lcm}}(S_k)$|⁠. On the other hand, |$d\mid{\mathrm{lcm}}(S_k)$|⁠. Hence |$d=\gcd(da_{j_k},{\mathrm{lcm}}(S_k))\in{\mathcal A}_{S_k}$|⁠. As |$d\not\in{\mathcal B}$|⁠, we see that |$d\in{\mathcal A}_{S_k}\setminus S_k$| for all |$k\in{\mathbb N}$|⁠. ■ Proposition 3.2. The following conditions are equivalent: (i) |$W$| is topologically regular, that is |$W=\overline{{\mathrm{int}}(W)}$|⁠. (ii) There are no |$d\in{\mathbb N}$| and no infinite pairwise coprime set |${\mathcal A}\subseteq{\mathbb N}\setminus\{1\}$| such that |$d\,{\mathcal A}\subseteq{\mathcal B}$|⁠. (iii) |${{\mathcal A}_\infty}=\emptyset$|⁠. (iv) |${\mathcal E}={\mathcal F}_{\mathcal B}$|⁠. □ Proof. The equivalence of |$(i)$| and |$(ii)$| follows from Proposition 3.1, that of |$(iii)$| and |$(iv)$| from Lemma 3.4. In view of Lemma 3.2, Proposition 3.1 finally implies the equivalence of |$(i)$| and |$(iii)$|⁠, too. ■ Lemma 3.5. |$\Delta({\mathbb Z})\cap\left(\overline{{\mathrm{int}}(W)}\setminus{\mathrm{int}}(W)\right)=\emptyset$|⁠. □ Proof. Assume |$\Delta(m)\in \overline{{\mathrm{int}}(W)}\setminus{\mathrm{int}}(W)$|⁠. Then for any |$S\subset {\mathcal B}$| there exists |$n_S\in{\mathbb Z}$| such that |$\Delta(n_S)\in U_S(\Delta(m))\cap {\mathrm{int}}(W)$|⁠. It means that for any |$S$| there exist: a finite set |$T_S\subset {\mathcal B}$|⁠, (we can assume that |$S\subset T_S$|⁠), |$b_S\in{\mathcal B}$| and |$n_S\in{\mathbb Z}$| such that (see Lemma 3.1 c)): |${\mathrm{lcm}}(S)|m-n_s$| (that is, |$\Delta(n_s)\in U_S(\Delta(m))$|⁠), |$\gcd(b_S,{\mathrm{lcm}}(T_S))$| does not divide |$n_S$| (⁠|$\Delta(n_S)$| is chosen to be an element of |$U_S(\Delta(m))\cap {\mathrm{int}}(W)$|⁠), |$\gcd(b_S,{\mathrm{lcm}}(T_S))|m$| (since |$\Delta(m)\notin {\mathrm{int}}(W)$|⁠). Then |$\gcd(b_S,{\mathrm{lcm}}(T_S))$| does not divide |${\mathrm{lcm}}(S)$|⁠. Let us iterate: |$S_0$| is arbitrary and |$S_{k+1}:=T_{S_k}$|⁠, |$c_k:={\mathrm{lcm}}(S_{k+1})$|⁠, |$d_k:=\gcd(b_{S_k},{\mathrm{lcm}}(S_{k+1}))$|⁠. We have: |$c_k|c_{k+1}$|⁠, |$d_k|m$|⁠, |$d_k|c_k$|⁠, |$d_k$| does not divide |$c_{k-1}$|⁠. Since |$d_k|m$| for every |$k$|⁠, the sequence |$({\mathrm{lcm}}(d_1,\ldots d_k))_k$| stabilizes on |${\mathrm{lcm}}(d_1,\ldots d_{k_0})$| for some |$k_0$|⁠, which means |$d_l$| divides |${\mathrm{lcm}}(d_1,\ldots d_{k_0})$|⁠, and consequently |$d_l$| divides |${\mathrm{lcm}}(c_1,\ldots,c_{k_0})=c_{k_0}$|⁠, for any |$l$|⁠, a contradiction. ■ For |$x\in\{0,1\}^{\mathbb Z}$| denote |${\mathrm{supp}} x:=\{n\in{\mathbb Z}: x(n)=1\}$|⁠. Following [4] we consider the set \begin{equation*} \begin{split} Y := & \left\{x\in\{0,1\}^{\mathbb Z}: |{\mathrm{supp}} x\text{ mod }b|=b-1\text{ for all }b\in{\mathcal B}\right\}\\ =& \big\{x\in\{0,1\}^{\mathbb Z}: \text{for all }b\in{\mathcal B}\text{ there is exactly one }r\in\{0,\dots,b-1\}\\ &\quad\text{ with }{\mathrm{supp}} x\cap(b{\mathbb Z}+r)=\emptyset\big\}. \end{split} \end{equation*} As |${\mathrm{supp}}\eta={\mathcal F}_{\mathcal B}$| is disjoint from |$b{\mathbb Z}$| for all |$b\in{\mathcal B}$|⁠, we have \begin{equation}\label{eq:eta-in-Y} \eta\in Y \Leftrightarrow \forall b\in{\mathcal B}\ \forall r\in\{1,\dots,b-1\}: {\mathcal F}_{\mathcal B}\cap (b{\mathbb Z}+r)\neq\emptyset. \end{equation} (27) Lemma 3.6. If |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠, then |$\eta\in Y$|⁠. □ Proof. For |$b\in{\mathcal B}$| and |$r\in\{0,\dots,b-1\}$| let |$V_b(r):=\{h\in H: {h_b=r\mod b}\}$| and observe that these sets are open and closed in |$H$|⁠. Hence |$\overline{\Delta({\mathbb Z})\cap V_b(r)\cap W}=V_b(r)\cap W$|⁠, because |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠. Suppose for a contradiction that |$\eta\not\in Y$|⁠. Then (27) implies that there are |$b\in{\mathcal B}$| and |$r\in\{0,\dots,b-1\}$| such that \begin{equation*} \Delta({\mathbb Z})\cap V_b(r)\cap W=\emptyset, \end{equation*} which implies that also |$V_b(r)\cap W=\emptyset$|⁠. Hence \begin{equation*} V_b(r)\subseteq W^c= \bigcup_{b'\in{\mathcal B}}V_{b'}(0), \end{equation*} and as |$V_b(r)$| is compact and the |$V_{b'}(0)$| are open, there is a finite |$S\subset{\mathcal B}$| such that \begin{equation*} V_b(r)\subseteq \bigcup_{b'\in S}V_{b'}(0). \end{equation*} In other words, whenever |$h_b=r$| for some |$h\in H$|⁠, then |$h_{b'}=0$| for some |$b'\in S$|⁠. Applied to any |$h=\Delta(n)$| this yields: \begin{equation*} n\in b{\mathbb Z}+r\;\Rightarrow\;n\in\bigcup_{b'\in S}b'{\mathbb Z}. \end{equation*} Since |$r$| is not divisible by |$b$|⁠, we can assume that |$b\notin S$|⁠. Let |$q=\gcd(b,r)$|⁠, |$\tilde{b}=b/q$|⁠, |$\tilde{r}=r/q$|⁠. Then |$q\,(\tilde{b}{\mathbb Z}+\tilde{r})=b{\mathbb Z}+r\subseteq{\mathcal M}_S$|⁠, so that |${\mathcal M}_{\tilde{b}{\mathbb Z}+\tilde{r}}\subseteq{\mathcal M}_{S'(q)}$| by Lemma 2.2. But |$d({\mathcal M}_{\tilde{b}{\mathbb Z}+\tilde{r}})=1$| by Dirichlet’s theorem, whereas |$d({\mathcal M}_{S'(q)})<1$|⁠, because |$S'(q)\subseteq\{1,\dots,\max S\}$| is finite {and |$1\notin S'(q)$|⁠.} (As |$q|b$| and |${\mathcal B}$| is primitive, |$q\notin S$|⁠, thus |$1\notin S'(q)$|⁠.) This is a contradiction. ■ Remark 3.1. Together with Theorem A this shows that |$\eta\in Y$| whenever |${\mathcal B}$| is taut. This implication was proved previously in [4, Corollary 4.32]. □ Lemma 3.6 provides the implication \begin{equation*} \overline{\Delta({\mathbb Z})\cap W}=W\Rightarrow \eta\in Y. \end{equation*} The reverse implication does not hold, as is shown by the next example. Example 3.1. Observe that for every |$k\in{\mathbb Z}$| there exists a prime divisor |$p_k$| of |$5+12k$| such that \begin{equation}\label{ex_1} p_k\neq 1\mod 12 \;\;\text{and}\;\;p_k\neq -1\mod 12. \end{equation} (28) Let $${\mathcal B}=\{4,6\}\cup\{p_k:k\in{\mathbb Z}\}.$$ Let us enumerate the elements of |${\mathcal B}$| as |$b_0,b_1,b_2,\ldots$| and |$b_0=4, b_1=6$|⁠. Observe that \begin{equation}\label{ex_2} 5+12{\mathbb Z}\subset{\mathcal M}_{{\mathcal B}}. \end{equation} (29) Since neither 2 nor 3 divides an element of the progression |$5+12{\mathbb Z}$|⁠, in view of (28) we see that |$1,2,3,11,22\in{\mathcal F}_{{\mathcal B}}$|⁠. It follows that \begin{equation}\label{ex_3} |{\mathrm{supp}}{\mathcal F}_{{\mathcal B}} \mod 4|=3\;\;\text{and} \;\; |{\mathrm{supp}}{\mathcal F}_{{\mathcal B}} \mod 6|=5. \end{equation} (30) We claim that \begin{equation}\label{ex_4} |{\mathrm{supp}}{\mathcal F}_{{\mathcal B}} \mod b_k|=b_k-1\;\text{for any} \; k\ge 2. \end{equation} (31) It is clear that |$\gcd(12,b_k)=1$| for any |$k\ge 2$|⁠. Let |$k\ge 2$| and take arbitrary |$r\in\{1,\ldots,b_k-1\}$|⁠. There exists |$r'\in{\mathbb Z}$| such that \begin{equation}\label{ex_5} \left\{\begin{array}{c} r'\equiv r\mod b_k\\ r'\equiv 1 \mod 12. \end{array}\right. \end{equation} (32) Then |$\gcd(12b_k,r')=1$| and, by Dirichlet’s theorem, there exists a prime number |$q$| of the form |$q=12b_kl+r'$| for some |$l\in{\mathbb Z}$|⁠. Since, by (32), |$q\equiv 1\mod 12$|⁠, |$q\in{\mathcal F}_{{\mathcal B}}$| by (28). Moreover, |$q\equiv r\mod b_k$| by (32). Thus the claim (31) follows. Clearly, (31) and (30) yield |$\eta\in Y$|⁠. We shall construct |$h\in W$| such that |$h\notin\overline{\Delta({\mathbb Z})\cap W}$|⁠. We denote |$S_k=\{b_0,b_1,\ldots b_k\}$|⁠. Inductively we construct a sequence |$(n_{S_k})$| of integers satisfying: (a) |$n_{S_1}=5$| (b) |${\mathrm{lcm}}(S_k)|n_{S_{k+1}}-n_{S_k}$| for |$k=1,2,\ldots$| (c) |$n_{S_k}\in{\mathcal F}_{S_k}$| for |$k=1,2,\ldots$| Assume that |$n_{S_1},\ldots,n_{S_k}$| have been constructed. If |$b_{k+1}$| does not divide |$n_{S_k}$|⁠, we set |$n_{S_{k+1}}=n_{S_k}$|⁠. Otherwise we set |$n_{S_{k+1}}=n_{S_k}+{\mathrm{lcm}}(S_k)$|⁠. The conditions a), b), c) follow easily by induction. Let $$ h=\lim_k\Delta(n_{S_k}). $$ Thanks to c), |$h\in W$|⁠. But $$ U_{S_1}(h)\cap\Delta({\mathbb Z})\cap W=U_{S_1}(\Delta(5))\cap\Delta({\mathbb Z})\cap W=\Delta(5+12{\mathbb Z})\cap W=\emptyset, $$ the last equality by (29). (Clearly, |$d({\mathcal M}_{{\mathcal B}})=1$| and |${\mathcal B}$| is not taut.) □ 3.2 Proof of Theorem B Lemma 3.7. If |${\mathcal B}$| is primitive and |$\eta$| is a Toeplitz sequence, then |${\mathcal B}$| is taut. (The authors are indebted to A. Bartnicka for pointing out and proving this lemma.) □ Proof. Suppose that |${\mathcal B}$| is not taut. Then there are |$c\in{\mathbb N}$| and a Behrend set |${\mathcal A}$| such that |$c{\mathcal A}\subseteq {\mathcal B}$|⁠. Hence \begin{equation}\label{eq:Aurelia} d({\mathcal M}_{\mathcal B}\cap c{\mathbb Z})=c^{-1}, \end{equation} (33) because |${\mathcal M}_{\mathcal A}$| has density one. As |${\mathcal B}$| is primitive, |$c$| must be |${\mathcal B}$|-free. So |$\eta(c)=1$|⁠, and (since |$\eta$| is Toeplitz) there exists |$m\in{\mathbb N}$| such that |$c+m{\mathbb Z}\subseteq {\mathcal F}_{\mathcal B}$|⁠. But then \begin{equation*} \underline{d}({\mathcal F}_{\mathcal B}\cap c{\mathbb Z}) \geqslant \underline{d}((c+m{\mathbb Z})\cap c{\mathbb Z}) = d({\mathrm{lcm}}(c,m){\mathbb Z}) ={\mathrm{lcm}}(c,m)^{-1}>0, \end{equation*} which contradics (33). ■ Lemma 3.8. Assume that |$\eta\in Y$|⁠. If |$\eta={\rm 1}\kern-0.24em{\rm I}_{{\mathcal F}_{\mathcal B}}$| is almost periodic (that is if the orbit closure of |$\eta$| is minimal), then |$X_\eta\subseteq Y$|⁠. □ Proof. Fix |$k\geq1$|⁠. Since |$\eta\in Y$|⁠, the support of |$\eta$| taken mod |$b_k$| misses exactly one residue class mod |$b_k$| (that is, it misses zero). Let |$B$| be a block on |$\eta$| such that its support mod |$b_k$| misses exactly one residue class mod |$b_k$|⁠. Since |$\eta$| is almost periodic, the block |$B$| appears on |$\eta$| with bounded gaps. It follows that if |$C$| is any sufficiently long block that appears on |$\eta$|⁠, its support misses exactly one residue class. Clearly this property passes to limits in the product topology, so each |$y=\lim S^{m_i}\eta$| is also in |$Y$|⁠. ■ In general, we can define a map |$\theta:Y\to\prod_{k\geq1}{\mathbb Z}/b_k{\mathbb Z}$| by setting \begin{equation*} \theta(y)=g=(g_k)_{k\geq1}\text{ iff } {\mathrm{supp}}\,\,y\cap(b_k{\mathbb Z}-g_k)=\emptyset\text{ for all }k\geq1. \end{equation*} Remark 2.51 in [4] tells us that \begin{equation*} \theta(Y\cap X_\eta)\subset H, \end{equation*} while Remark 2.52 says that |$\theta$| is continuous. Corollary 3.1. By the definitions of |$\varphi$| and |$\theta$|⁠, we have |$\theta\circ\varphi(h)=h$| provided |$\varphi(h)\in Y$|⁠. In particular, |$\theta(\eta)=0$| and |$\theta$| is continuous at |$\eta$|⁠. Moreover, |$\theta$| is equivariant. □ For any map |$\psi:X\to Y$| denote by |$C_\psi\subseteq X$| the set of continuity points of this map. Lemma 3.9. Let |$(X,S)$| and |$(Y,T)$| be compact dynamical systems and assume that |$(X,S)$| is minimal. Let |$\psi:X\to Y$| be a map satisfying |$\psi\circ S=T\circ\psi$|⁠. Then |$\overline{\psi(C_\psi)}$| is a minimal subset of |$Y$|⁠. □ Proof. Denote by |$Z:=\overline{\{(x,\psi(x)): x\in X\}}$| the closure of the graph of |$\psi$| and note that a fibre |$Z_x=\{y:(x,y)\in Z\}$| is a singleton, if and only if |$x\in C_\psi$|⁠. Let |$Z_0:=\overline{\{(x,\psi(x)): x\in C_\psi\}}$|⁠. We claim that |$Z_0\subseteq A$| whenever |$A$| is a non-empty closed |$S\times T$|-invariant subset of |$Z$|⁠. Indeed, |$\pi_X(A)$| is a non-empty closed |$S$|-invariant subset of |$X$|⁠, so |$\pi_X(A)=X$| by minimality of |$(X,S)$|⁠. In particular, |$C_\psi\subseteq\pi_X(A)$|⁠. As all |$A_x\subseteq Z_x$| with |$x\in C_\psi$| are singletons, |$\{(x,\psi(x)):x\in C_\psi\}\subseteq A$|⁠. Hence also |$Z_0\subseteq A$|⁠. This shows that |$Z_0$| is a minimal subset of |$X\times Y$| (and, by the way, that it is the only minimal subset of |$Z$|⁠). It follows that |$\pi_Y(Z_0)$| is a minimal subset of |$Y$|⁠, and so it remains to show that |$\psi(C_\psi)\subseteq \pi_Y(Z_0)$|⁠. But, for |$x\in C_\psi$|⁠, |$(x,\psi(x))\in Z_0$|⁠, and so |$\psi(x)\in\pi_Y(Z_0)$|⁠. ■ Denote by |$C_\varphi$| the set of all points in |$H$| at which |$\varphi:H\to\{0,1\}^{\mathbb Z}$| is continuous. Lemma 3.10. (a) |$C_\varphi=\left\{h\in H:\ (h+\Delta({\mathbb Z}))\cap\partial W=\emptyset\right\}$|⁠. (b) |$C_\varphi+\Delta(1)=C_\varphi$|⁠. (c) |$\overline{\varphi(C_\varphi)}$| is the unique minimal subset |$M$|⁠. □ Proof. (a) This is proved by direct inspection, see for example, [13, Lemma 6.1]. (b) This is obvious. (c) This follows from Lemma 3.9. ■ Proof of Theorem B. We start with a list of implications, which, when suitably combined, prove the assertions (a) - (e) of Theorem B. Most of these implications can be proved without assuming that |${\mathcal B}$| is primitive and that |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠. Therefore we indicate explicitly, for which implications we use these extra assumptions. Proof of the equivalence of B1 – B4: These equivalences follow from Proposition 3.2. Proof of B1 |$\Rightarrow$| B6: Observe first that |$0\in H$| belongs to |$C_\varphi$| if and only if |$\Delta({\mathbb Z})\cap\partial W=\emptyset$|⁠, see Lemma 3.10. But |$\Delta({\mathbb Z})\cap\partial W=\Delta({\mathbb Z})\cap\left(\overline{{\mathrm{int}}(W)}\setminus{\mathrm{int}}(W)\right)$| in view of B1, and this intersection is empty by Lemma 3.5. As |${\mathrm{int}}(W)\neq\emptyset$| and as |$H=\overline{\Delta({\mathbb Z})}$|⁠, |$\Delta({\mathbb Z})\cap W\neq\emptyset$| and hence |$\varphi(0)\neq(\dots,0,0,0,\dots)$|⁠. Proof of B6 |$\Rightarrow$| B5: Let |${\mathcal B}=\{b_1,b_2,\ldots\}$| and assume (B6) that |$0\in C_\varphi$|⁠, that is |$\Delta({\mathbb Z})\cap\partial W=\emptyset$|⁠, and |$\eta\neq(\dots,0,0,0,\dots)$|⁠. Now, take |$n\in{\mathbb Z}$|⁠. Either |$n\in{\mathcal M}_{{\mathcal B}}$| - then |$\eta(n)=0$|⁠, so |$b_s\mid n$| for some |$s\geq1$| and |$\eta(n+jb_s)=0$| for each |$j\in{\mathbb Z}$|⁠. Or |$n\in {\mathcal F}_{{\mathcal B}}$|⁠, that is |$\Delta(n)\in W$|⁠. As |$\Delta({\mathbb Z})\cap\partial W=\emptyset$| by assumption, this implies |$\Delta(n)\in{\mathrm{int}}(W)$|⁠, so that |$n\in{\mathcal E}=\bigcup_{S\subset{\mathcal B}}{\mathcal F}_{{\mathcal A}_S}$| by Lemma 3.1. Hence there is a finite subset |$S\subset{\mathcal B}$| such that |$n\in{\mathcal F}_{{\mathcal A}_S}$|⁠. As |${\mathrm{lcm}}({\mathcal A}_S)={\mathrm{lcm}}(S)$|⁠, this implies \begin{equation*} n+{\mathrm{lcm}}(S)\,{\mathbb Z}\subseteq{\mathcal F}_{{\mathcal A}_S}\subseteq{\mathcal E}\subseteq{\mathcal F}_{\mathcal B}. \end{equation*} Hence |$\eta(n+j{\mathrm{lcm}}(S))=1$| for each |$j\in{\mathbb Z}$|⁠. This proves that |$\eta$| is a Toeplitz sequence different from |$(\dots,0,0,0,\dots)$|⁠. Proof of B5 |$\Rightarrow$| B1: Assume that |$\eta$| is a Toeplitz sequence. Then |${\mathcal B}$| is taut by Lemma 3.7, hence |$\overline{\Delta({\mathbb Z})\cap W}=W$| by Theorem A. Now B1 follows from the chain of the next three implications. Proof of B5 |$\Rightarrow$| B8: Each Toeplitz sequence is almost periodic [8], [12, Theorem 4], that is its orbit closure is minimal. Proof of B8 |$\Rightarrow$| B7: If |$X_\eta=M$|⁠, then |$\eta\in M$|⁠, and |$\eta\neq(\dots,0,0,0,\dots)$|⁠, because otherwise the minimality of |$X_\eta$| implies |$X_\eta=\{(\dots,0,0,0,\dots)\}$|⁠, contradicting |${\mathrm{card}}(X_\eta)>1$|⁠. Proof of B7 |$\Rightarrow$| B1 (assuming that |$\overline{\Delta({\mathbb Z})\cap W}=W$|⁠): Assume that |$(\dots,0,0,0,\dots)\neq\eta\in M=\overline{\varphi(C_\varphi)}$|⁠. Then |$M=X_\eta\subseteq Y$| by Lemma 3.8, and there is a sequence |$h_1,h_2,\dots\in C_\varphi$| such that |$\eta=\lim_{i\to\infty}\varphi(h_i)$|⁠. Consider |$n\in{\mathbb Z}$| with |$\Delta(n)\in W$|⁠, that is such that |$\eta(n)=1$|⁠. In particular |$\eta=\varphi(0)\neq(\dots,0,0,0,\dots)$|⁠. Corollary 3.1 implies |$\lim_{i\to\infty}h_i=\lim_{i\to\infty}\theta(\varphi(h_i))=\theta(\eta)=0$|⁠. Then |$1=\eta(n)=\lim_{i\to\infty}\varphi(h_i)(n)$|⁠, that is |$h_i+\Delta(n)\in W$| for all sufficiently large |$i$|⁠. As |$h_i\in C_\varphi$|⁠, we have |$(h_i+\Delta({\mathbb Z}))\cap\partial W=\emptyset$| (Lemma 3.10). Hence, |$h_i+\Delta(n)\in {\mathrm{int}}(W)$| for all sufficiently large |$i$|⁠, {which} implies that |$\Delta(n)=\lim_{i\to\infty}h_i+\Delta(n)\in\overline{{\mathrm{int}}(W)}$|⁠. This proves that |$\Delta({\mathbb Z})\cap W\subseteq\overline{{\mathrm{int}}(W)}$|⁠. Hence |$W=\overline{\Delta({\mathbb Z})\cap W}\subseteq\overline{{\mathrm{int}}(W)}$|⁠, that is |$W$| is topologically regular. Proof of B7 |$\Rightarrow$| B8: As |$\eta\in M$|⁠, also |$X_\eta\subseteq M$|⁠, and hence |$X_\eta= M$|⁠. As |$\eta\neq(\dots,0,0,0,\dots)$|⁠, |$X_\eta$| contains no fixed point. Hence |${\mathrm{card}}(X_\eta)>1$|⁠. Proof of B1 |$\Rightarrow$| B9 (assuming that |${\mathcal B}$| is primitive): The window |$W$| is aperiodic because of Proposition 5.1, and it is topologically regular by B1. As B1 |$\Rightarrow$| B8, |$X_\eta$| is minimal. Therefore, Corollary 1a) of [13], together with Lemmas 4.5 and 4.6 of the same reference, implies B9. Proof of B9 |$\Rightarrow$| B8: This is trivial. ■ Proposition 3.3. Assume that the window |$W$| is topologically regular. Then |$X_\eta\subseteq Y$|⁠. □ Proof. We start proving that |$\eta\in Y$|⁠. Assume the contrary, that is, there are |$b_0\in{\mathcal B}$| and |$r\in\{1,\ldots,b_0-1\}$| such that \begin{equation}\label{eta_in_Y1} r+b_0{\mathbb Z}\subset{\mathcal M}_{{\mathcal B}}. \end{equation} (34) Let |$a=\gcd(r,b_0)$| and |$r'=r/a$|⁠, |$b_0'=b_0/a$|⁠. (34) yields that for any |$k\in {\mathbb N}$| there exists |$b_k\in{\mathcal B}$| such that \begin{equation*} b_k\mid a(r'+kb_0'). \end{equation*} Let |$J=\{k\in{\mathbb N}:r'+kb_0'\;\text{is prime}\}$|⁠. By Dirichlet’s theorem the set |$J$| is infinite. As |${\mathcal B}$| is primitive, |$b_k$| does not divide |$a=\gcd(r,b_0)$|⁠. Hence, $$ b_k=\gcd(a,b_k)\,(r'+kb_0') $$ for any |$k\in J$|⁠. Since |$a$| has only finitely many divisors, there exists a divisor |$a'$| such that $$ b_k=a'(r'+kb_0') $$ for infinitely many |$k\in J$|⁠. Thus we obtain a contradiction with the condition (B4) of Theorem B, which is equivalent to (B1) |$W=\overline{{\mathrm{int}} W}$|⁠. Thus |$\eta\in Y$|⁠. Assume now that |$x\in X_\eta$| and let |$b\in{\mathcal B}$|⁠. As |$\eta\in Y$|⁠, there is |$N_b\in{\mathbb N}$| such that |${\mathrm{card}}\left({\mathrm{supp}}(\eta|_{[0:N_b]})\mod b\right)=b-1$|⁠. As |$X_\eta$| is minimal by (B8) of Theorem B, there is |$n\in{\mathbb N}$| such that |${\mathrm{supp}}(x|_{[n:n+N_b]})={\mathrm{supp}}(\eta|_{[0:N_b]})$|⁠. Hence \[ {\mathrm{card}}\left({\mathrm{supp}}(x)\mod b\right)\geqslant {\mathrm{card}}\left({\mathrm{supp}}(x|_{[n:n+N_b]})\mod b\right) \geqslant{\mathrm{card}}\left({\mathrm{supp}}(\eta|_{[0:N_b]})\mod b\right)= b-1, \] so that |$x\in Y$|⁠, because |${\mathrm{card}}\left({\mathrm{supp}}(x)\mod b\right)\leqslant b-1$| for all |$x\in X_\eta$|⁠, see Remark 1.5. ■ 3.3 Proof of Theorem C The equivalence of C1, C2 and C3 follows from Lemma 3.1. If C1 holds, that is if |${\mathrm{int}}(W)=\emptyset$|⁠, then |$\varphi(C_\varphi)=\{(\dots,0,0,0,\dots)\}$| is a shift invariant set [13, Proposition 3.3d with Remark 3.2b], so that |$M=\overline{\varphi(C_\varphi)}=\{(\dots,0,0,0,\dots)\}$|⁠. This is C6, and Theorem 3.8 in [4] shows that C4, C5, C6 and C7 are all equivalent. We finish by proving C5 |$\Rightarrow$| C3: Consider any finite |$S\subset {\mathcal B}$|⁠. As |${\mathcal B}\subseteq{\mathcal M}_{{\mathcal A}_S}$| by definition of the set |${\mathcal A}_S$|⁠, C5 implies that |$1\in{\mathcal A}_S$|⁠. 4 The Sequence |$\boldsymbol{{\mathcal B}}$| and Haar Measure 4.1 Measure and density Lemma 4.1. |$m_H(W)=1-\underline d({\mathcal M}_{\mathcal B})=\bar d({\mathcal F}_{\mathcal B})$|⁠. □ Proof. For |$S\subset{\mathcal B}$| denote by |${\mathcal U}_S$| the family of all sets |$U_S(\Delta(n))$| that are contained in |$W^c$| and by |$\cup{\mathcal U}_S$| the union of these sets. Then \begin{align*} m_H(W^c) &= \sup_S m_H(\cup{\mathcal U}_S)\\ & = \sup_S \frac{\#{\mathcal U}_S}{{\mathrm{lcm}}(S)} \geqslant \sup_S \frac{\#({\mathcal M}_S\cap\{1,\dots,{\mathrm{lcm}}(S)\})}{{\mathrm{lcm}}(S)} = \sup_S d({\mathcal M}_S) =\underline d({\mathcal M}_{\mathcal B}) \end{align*} by Lemma 2.5a, and similarly \begin{equation*} m_H(W^c) \leqslant \sup_S \frac{\#({\mathcal M}_{{\mathcal B}\cap{\mathcal A}_S}\cap\{1,\dots,{\mathrm{lcm}}(S)\})}{{\mathrm{lcm}}(S)} = \sup_S d({\mathcal M}_{{\mathcal B}\cap{\mathcal A}_S}) \leqslant \underline d({\mathcal M}_{\mathcal B}) \end{equation*} by Lemma 2.5b. ■ Corollary 4.1. [4, Theorem 4.1]|${\mathcal B}$| is a Besicovich sequence if and only if the sequence |${\rm 1}\kern-0.24em{\rm I}_{{\mathcal F}_{{\mathcal B}}}$| is generic for the Mirsky measure. □ Proof. If |${\mathcal B}$| is Besicovich, then |$d({\mathcal F}_{\mathcal B})=m_H(W)$|⁠, so that |${\mathcal F}_{\mathcal B}$| has maximal density. Hence it is generic for the Mirsky measure, see [13, Theorem 5b]. On the other hand, if |${\mathcal F}_{\mathcal B}$| is generic for (any) measure, then its frequency of ones converges in particular, which means that its asymptotic density exists. ■ Lemma 4.2. |$m_H({\mathrm{int}}(W))=\sup_S d({\mathcal F}_{{\mathcal A}_S})\leqslant \underline d({\mathcal E})$|⁠. □ Proof. For |$S\subset{\mathcal B}$| denote by |${\mathcal U}^o_S$| the family of all sets |$U_S(\Delta(n))$| that are contained in |${\mathrm{int}}(W)$| and by |$\cup{\mathcal U}^o_S$| the union of these sets. Recall from Lemma 3.1a that |$\#{\mathcal U}^o_S=\#({\mathcal F}_{{\mathcal A}_S}\cap\{1,\dots,{\mathrm{lcm}}(S)\})$|⁠. Then \[ m_H({\mathrm{int}}(W)) = \sup_S m_H(\cup{\mathcal U}^o_S) = \sup_S \frac{\#{\mathcal U}^o_S}{{\mathrm{lcm}}(S)} = \sup_S \frac{\#({\mathcal F}_{{\mathcal A}_S}\cap\{1,\dots,{\mathrm{lcm}}(S)\})}{{\mathrm{lcm}}(S)} = \sup_S d({\mathcal F}_{{\mathcal A}_S}). \] ■ Lemma 4.3. |$m_H(\partial W)=\inf_S{\underline d}({\mathcal M}_{{\mathcal A}_S}\setminus{\mathcal M}_{\mathcal B})\leqslant \inf_S d({\mathcal M}_{{\mathcal A}_S\setminus{\mathcal B}})$|⁠. □ Proof. We have \[ \begin{split} m_H(\partial W) &= m_H(W)-m_H({\mathrm{int}}(W)) = {\underline d}({\mathcal F}_{\mathcal B})-\sup_S d({\mathcal F}_{{\mathcal A}_S}) =\inf_S\left({\underline d}({\mathcal F}_{\mathcal B})-d({\mathcal F}_{{\mathcal A}_S})\right)\\ &= \inf_S{\underline d}({\mathcal M}_{{\mathcal A}_S}\setminus{\mathcal M}_{\mathcal B}). \end{split} \] ■ 4.2 Regular Toeplitz sequences Let |${\mathcal B}=\{b_1,b_2,\ldots\}$|⁠. For each |$k\geq1$|⁠, consider the sequence $$ b_1,\ldots,b_k,c^{(k)}_{k+1},c^{(k)}_{k+2},\ldots,$$ where $$ c^{(k)}_{k+i}:={\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+i}),\;i\geq1.$$ Then: \begin{gather} c^{(k)}_{k+i}|{\rm lcm}(b_1,\ldots,b_k), \mbox{whence $\{c^{(k)}_{k+i}:\:i\geq1\}$ is finite},\nonumber\\ \mathscr{A}_{\{b_1,\ldots,b_k\}}=\{b_1,\ldots,b_k\}\cup\{c^{(k)}_{k+i}:\:i\geq1\},\nonumber\\ c^{(k)}_{k+1}|b_{k+1},\nonumber\\ c^{(k)}_{k+1+i}|c^{(k+1)}_{k+1+i},\;\text{for each }i\geq1.\label{cirm4} \end{gather} (35) Moreover, following Lemma 2.5c, there is an increasing sequence |$(k_n)$| such that \begin{equation}\label{cirm0} {\mathcal B}\cap \mathscr{A}_{\{b_1,\ldots,b_{k_n}\}}=\{b_1,\ldots,b_{k_n}\}. \end{equation} (36) We assume that |$W\subset H$| is topologically regular, so by Remark 1.3, |$\eta={\rm 1}\kern-0.24em{\rm I}_{{\mathcal F}_{{\mathcal B}}}$| is a Toeplitz sequence. We set |$s_k:={\rm lcm}(b_1,\ldots,b_k)$| and would like now to examine the sequence |$(s_k)$| as a periodic structure of |$\eta$|⁠. More precisely, we would like to see for how many |$n\in[1,s_k]$|⁠, we have |$\eta(n)=\eta(n+js_k)$| for each |$j\in{\mathbb Z}$|⁠. We call any such |$n$| to be “good”. Now, if |$n\in {\mathcal F}_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$|⁠, then |$n+s_k{\mathbb Z}\subset {\mathcal F}_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$|⁠, so |$n$| is good. Otherwise, |$n\in {\mathcal M}_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}$|⁠. Then either |$n\in {\mathcal M}_{\{b_1,\ldots,b_k\}}$| and then clearly |$\eta(n+js_k)=0$| for each |$j\in{\mathbb Z}$|⁠, so again |$n$| is good, or $$n\in {\mathcal M}_{\{c^{(k)}_{{k}+i}:\:i\geq1\}}\setminus{\mathcal M}_{\{b_1,\ldots,b_k\}}.$$ Only for such |$n$|⁠, we are not sure that |$n$| is good. Moreover, note that in view of (35), we have \begin{equation*}\label{cirm5} {\mathcal M}_{\{c^{(k+1)}_{k+1+i}:\:i\geq1\}}\subset {\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}},\end{equation*} so the sequence |$(d({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}}))_k$| is decreasing, and so is the sequence |$(\overline{d}({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}}\setminus{\mathcal M}_{{\mathcal B}}))_{k}$|⁠. Therefore, by taking into account (36), the infimum of this sequence is equal to the liminf, in fact to the limit and we have \begin{equation}\label{cirm6} { \inf_{S\subset {\mathcal B}}\overline{d}({\mathcal M}_{\mathscr{A}_S}\setminus {\mathcal M}_{{\mathcal B}})=\liminf_{k\to\infty}\overline{d}({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}}\setminus{\mathcal M}_{\{b_1,\ldots,b_k\}}). } \end{equation} (37) Definition 4.1. Let |$\eta={\rm 1}\kern-0.24em{\rm I}_{{\mathcal F}_{{\mathcal B}}}$| be a Toeplitz sequence. It is a regular Toeplitz sequence for the periodic structure |$(s_k)$|⁠, |$s_k={\rm lcm}(b_1,\ldots,b_k)$|⁠, if the |$\liminf$| in (37) is zero. □ Now, using Lemma 4.3, the identity in (37) shows the following result. Proposition 4.1. If |$W$| is topologically regular, then |$\eta={\rm 1}\kern-0.24em{\rm I}_{{\mathcal F}_{{\mathcal B}}}$| is a regular Toeplitz sequence for the periodic structure |$(s_k)$|⁠, |$s_k={\rm lcm}(b_1,\ldots,b_k)$|⁠, if and only if |$m_H(\partial W)\,{=}\,0$|⁠. □ Example 4.1. Assume that |$\{b'_k:\:k\geq1\}$| is a coprime set of odd numbers and let |$b_k=2^kb'_k$|⁠. Then |$c^{(k)}_{k+i}=2^k$| for each |$i\geq1$|⁠. Hence, we have even |$d({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}})\to 0$|⁠, in particular |$\eta$| is a regular Toeplitz sequence for the periodic structure |$(s_k)$| with |$s_k=2^kb_1'\cdots b_k'$|⁠. This example comes from [4]. □ We will now show that we can obtain Toeplitz sequences also in case |$m_H(\partial W)\,{>}\,0$|⁠. Example 4.2. We will construct |${\mathcal B}=\{b_1,b_2,\ldots\}$| such that this set is thin (hence taut) and that |$\lim_{k\to\infty}\inf(\{c^{(k)}_{k+i}:\:i\geq1\}\setminus\{b_1,\ldots,b_k\})=\infty$|⁠, which, by the equivalence of the conditions |$(i)$| and |$(iii)$| in Proposition 3.2, implies that |$W$| is topologically regular (and hence |$\eta$| is Toeplitz by Theorem B). Let |$\delta_k>0$| and |$\sum_{k\geq1}\delta_k < 1/16$|⁠. We start with |$b_1=2^3$| and set |$c^{(1)}_{1+i}=2$| for each |$i\geq1$|⁠. Suppose that a sequence $$ b_1,\ldots, b_k, c^{(k)}_{k+1},c^{(k)}_{k+2},\ldots$$ has been defined. We require that this sequence satisfies: \begin{gather}\label{cirm7} c^{(k)}_{k+i}|{\rm lcm}(b_1,\ldots,b_k),\;i\geq1,\\ \end{gather} (38) \begin{gather} \label{cirm8} c^{(k)}_{k+i}\notin \{b_1,\ldots,b_k\}, \;i\geq1,\\ \end{gather} (39) \begin{gather} \label{cirm9} \mbox{for each $i\geq1$, $|\{j\geq1:\:c^{(k)}_{k+j}=c^{(k)}_{k+i}\}|=+\infty$}. \end{gather} (40) We will now show how to define |$c^{(k+1)}_{k+2}$|⁠, |$c^{(k+1)}_{k+3}$|⁠, |$\ldots$| and then |$b_{k+1}$|⁠. Recall an elementary lemma. Lemma 4.4. Let |$F_1, F_2$| be finite sets of natural numbers such that |${\rm gcd}(f_1,f_2)=1$| for each |$f_i\in F_i$|⁠, |$i=1,2$|⁠. Then |$d({\mathcal M}_{F_1\cdot F_2})=d({\mathcal M}_{F_1}\cap{\mathcal M}_{F_2})=d({\mathcal M}_{F_1})d({\mathcal M}_{F_2})$|⁠. □ Given a set |$A$| of natural numbers we denote by |${\rm spec}\,A$| the spectrum of |$A$|⁠, that is, the set of the primes dividing an element of |$A$|⁠. Choose |$P\subset \mathscr{P}\setminus{\rm spec}\,\{b_1,\ldots,b_k\}$|⁠, so that (by Lemma 4.4) \begin{equation}\label{cirm10} d({\mathcal M}_{P\cdot\{c^{(k)}_{k+i}:\:i\geq1\}})\geq d({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}})-\delta_k.\end{equation} (41) In view of (40), $$ \{i\geq2:\:c^{(k)}_{k+i}=c^{(k)}_{k+1}\}=\{r_1,r_2,\ldots\}.$$ Let |$P=\{q_1,\ldots,q_t\}$|⁠. Then set $$ c^{(k+1)}_{k+{r_{1+tj}}}:=c^{(k)}_{k+1}q_1,c^{(k+1)}_{k+r_{2+tj}}:=c^{(k)}_{k+1}q_2,\ldots, c^{(k+1)}_{k+r_{t+tj}}:=c^{(k)}_{k+1}q_t$$ for each |$j=0,1,\ldots$| If |$2\notin \{i\geq2:\:c^{(k)}_{k+i}=c^{(k)}_{k+1}\}$| then repeat the same construction with the set |$\{i\geq2:\: c^{(k)}_{k+i}=c^{(k)}_{k+2}\}$|⁠. Since (by (38)) the set |$\{c^{(k)}_{k+i}:\:i\geq1\}$| is finite, our construction of the sequence |$(c^{(k+1)}_{k+1+i})_i$| is done in finitely many steps. Finally, we set |$b_{k+1}:=c^{(k)}_{k+1}\prod_{q\in P}q$| (or, if needed, |$b_{k+1}:=c^{(k)}_{k+1}\prod_{q\in P}q^{\alpha_{k+1}}$| for any |$\alpha_{k+1}\in{\mathbb N}$|⁠). Note that $$ {\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+1})=c^{(k)}_{k+1}$$ since |$P\cap\{b_1,\ldots,b_k\}=\emptyset$|⁠. More than that, by the construction, we also have $$ {\rm gcd}({\rm lcm}(b_1,\ldots,b_k),b_{k+i})=c^{(k)}_{k+i}\text{ for each }i\geq1.$$ Moreover, it is not hard to see that the new sequence $$ b_1,\ldots,b_k,b_{k+1},c^{(k+1)}_{k+2},c^{(k+1)}_{k+3},\ldots$$ satisfies (38)–(40). Furthermore, |${\mathcal B}=\{b_1,b_2,\ldots\}$| satisfies the other requirements mentioned at the beginning of the construction so that |$\eta$| is a Toeplitz sequence and |$W$| is topologically regular. In our construction |$d({\mathcal M}_{\{c^{(1)}_{1+i}:\:i\geq1\}})=1/2$|⁠. Moreover, by (41) $$ d({\mathcal M}_{\{c^{(k)}_{k+i}:\:i\geq1\}})\geq d({\mathcal M}_{\{c^{(1)}_{1+i}:\:i\geq1\}})-\sum_{j=1}^k{\delta_j}\geq \frac14$$ for each |$k\geq1$|⁠. Finally notice that |$d({\mathcal M}_{{\mathcal B}})\leq \sum_{k\geq1}1/b_k$|⁠, which (by construction) can be made smaller than 1/8. It follows that |$\lim_{k\to\infty}\overline{d}({\mathcal M}_{\mathscr{A}_{\{b_1,\ldots,b_k\}}}\setminus {\mathcal M}_{{\mathcal B}})>0$|⁠, whence |$m_H(\partial W)>0$|⁠. Note that |$\eta$| is irregular by Proposition 4.1. □ 5 The Maximal Equicontinuous Factor of |$\boldsymbol{X_\eta}$| 5.1 The period groups of |$W$| and of |${\mathrm{int}}(W)$| Given a subset |$A\subseteq H$|⁠, denote by \begin{equation*} H_A:=\left\{h\in H: W+h=W\right\} \end{equation*} the period group of |$A$|⁠. The set |$A\subseteq H$| is topologically aperiodic, if |$H_A=\{0\}$|⁠. The following simple observations are proved in [14, Lemma 6.1]: |$H_A\subseteq H_{\bar A}\cap H_{{\mathrm{int}}(A)}$|⁠. If |$A$| is closed, then |$H_{{\mathrm{int}}(A)}=H_{\overline{{\mathrm{int}}(A)}}$| is closed. Proposition 5.1. Assume that |${\mathcal B}$| is primitive. Then the window |$W$| is topologically aperiodic. □ Proof. Suppose that |$h=(h_b)_{b\in{\mathcal B}}\neq0$| and \begin{equation}\label{ape4} W+h=W. \end{equation} (42) Since |$h\neq0$|⁠, there is |$b\in{\mathcal B}$| such that |$b$| does not divide |$h_b$|⁠. Let |$n:=\gcd(b,h_b)$|⁠. Then |$n\in{\mathcal F}_{{\mathcal B}}$|⁠, as otherwise there exists |$b'\in {\mathcal B}$| such that |$b'\mid n$|⁠; but then |$b'\mid b$| a contradiction (⁠|${\mathcal B}$| is assumed to be primitive). Hence |$\Delta(n)\in W$|⁠. There are |$x,y\in{\mathbb Z}$| such that |$n=xh_b+yb$|⁠, whence |$b\mid n-xh_b$|⁠. It follows that |$\Delta(n)-xh\notin W$|⁠, a contradiction with (42). ■ If |$W$| is topologically regular, then clearly |${\mathrm{int}}(W)$| is topologically aperiodic, as well. Otherwise |$H_{{\mathrm{int}}(W)}$| may be non-trivial, as we will see in the course of this section. Recall from (1) that for any set |$A\subseteq{\mathbb N}$|⁠, \begin{equation*} {\mathcal M}_A = {\mathcal M}_{A^{prim}}. \end{equation*} If |$A^{prim}$| is finite, then |${\mathcal M}_A$| is a union of finitely many arithmetic progressions. Let |$c_A$| denote the period of |${\mathcal M}_A$|⁠, that is, the least natural number such that |$c_A+{\mathcal M}_A={\mathcal M}_A$|⁠. Lemma 5.1. Assume that |$A,B\subset{\mathbb N}$| are finite (a) If |$c+{\mathcal M}_A={\mathcal M}_A$| for some |$c\in{\mathbb N}$|⁠, then |${\mathrm{lcm}}(A^{prim})\mid c$|⁠. (b) |$c_A={\mathrm{lcm}}(A^{prim})$|⁠. (c) If |$A\subset{\mathcal M}_B$|⁠, then |$c_B\mid c_A$|⁠. □ Proof. (a) For any |$a\in A^{prim}$| we have |$a+c{\mathbb Z}\subseteq{\mathcal M}_{A^{prim}}$|⁠, from which it follows that there exists |$a'\in A^{prim}$| such that |$a'\mid \gcd(a,c)$|⁠. But, as |$A^{prim}$| is primitive, that means that |$a'=a$| and |$a\mid c$|⁠. We conclude that |${\mathrm{lcm}}(A^{prim})\mid c$|⁠. (b) Clearly |${\mathcal M}_A+{\mathrm{lcm}}(A^{prim})={\mathcal M}_A$|⁠, thus |$c_A\mid {\mathrm{lcm}}(A^{prim})$| and the assertion follows from (a). (c) If |$A\subset{\mathcal M}_B$| then |$A^{prim}\subset {\mathcal M}_{B^{prim}}$|⁠, hence |${\mathrm{lcm}}(B^{prim})\mid {\mathrm{lcm}}(A^{prim})$|⁠, and we finish by (b). ■ Lemma 5.2. Assume that |$S\subseteq S'$| are finite subsets of |${\mathcal B}$|⁠, then |${\mathcal A}_S=\{\gcd(a,{\mathrm{lcm}}(S)):a\in{\mathcal A}_{S'}\}$|⁠. □ Proof. Since |${\mathrm{lcm}}(S)\mid {\mathrm{lcm}}(S')$|⁠, |$\gcd(b,{\mathrm{lcm}}(S))=\gcd(\gcd(b,{\mathrm{lcm}}(S')),{\mathrm{lcm}}(S))$| for any |$b\in{\mathcal B}$| and the assertion follows. ■ Let |$S_1\subset S_2\subset\ldots\subset S_k\subset\ldots$| be a filtration of |${\mathcal B}$| with finite sets and denote \begin{equation*} s_k:={\mathrm{lcm}}(S_k),\; c_k:=c_{{\mathcal A}_{S_k}}. \end{equation*} By Lemma 5.1 c) we have |$c_l\mid c_{l+1}$| for any |$l$|⁠. It follows that, for any |$k$|⁠, the sequence |$(\gcd(s_k,c_l))_{l\ge 1}$| stabilizes on a divisor |$d_k$| of |$s_k$|⁠. Clearly, since |$c_k\mid s_k$|⁠, \begin{equation}\label{nowy_1G} c_k\mid d_k\mid s_k. \end{equation} (43) Observe that, \begin{equation}\label{nowy_2G} d_k=\gcd(s_k,d_{k+1}). \end{equation} (44) Indeed, there is |$l_0\in{\mathbb N}$| such that |$d_{k+1}=\gcd(s_{k+1},c_l)$| for all |$l>l_0$|⁠. Since |$s_k\mid s_{k+1}$|⁠, we get $$ \gcd(s_k,c_l)=\gcd(s_k,\gcd(s_{k+1},c_l)). $$ It follows that |$\gcd(s_k,c_l)=\gcd(s_k,d_{k+1})$| for |$l>l_0$|⁠, and (44) follows. By applying (44) we prove by induction that \begin{equation}\label{nowy_3G} d_k=\gcd(s_k,d_{k+j}). \end{equation} (45) for |$j\ge 0$|⁠. Lemma 5.3. Let |$(n_k)_{k\in{\mathbb N}}$| be a sequence of integers. The following are equivalent: \begin{equation}\label{eq:equiv1S} \forall k\in{\mathbb N}:\ c_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k, \end{equation} (46) and \begin{equation}\label{eq:equiv2} \forall k\in{\mathbb N}:\ d_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k. \end{equation} (47) □ Proof. As |$c_{k}\mid d_{k}$|⁠, (47) implies (46). Conversely, assume that (46) holds. We show inductively that for all |$j\geqslant0$| \begin{equation}\label{eq:inductive} \forall k\in{\mathbb N}:\ \gcd(s_{k},c_{k+j})\mid n_{k}, \end{equation} (48) and this implies (47) immediately. For |$j=0$|⁠, (48) follows from (46), because |$c_{k}\mid s_{k}$|⁠. So suppose that (48) holds for some |$j\geqslant0$|⁠. Then \[ \begin{split} n_{k+1}&=0\;\mod \gcd(s_{k+1},c_{k+1+j})\;\text{and}\\ n_{k+1}&=n_{k}\mod s_k. \end{split} \] Hence |$n_k=0\mod \gcd(s_k,s_{k+1},c_{k+1+j})=\gcd(s_k,c_{k+j+1})$| for all |$k\in{\mathbb N}$|⁠, that is (48) for |$j+1$|⁠. ■ Recall that |$H_{{\mathrm{int}}(W)}=\{h\in H:{\mathrm{int}}(W)+h={\mathrm{int}}(W)\}$| denotes the period group of |${\mathrm{int}}(W)$|⁠. Proposition 5.2. (a) |$h\in H_{{\mathrm{int}}(W)}$| if and only if |$h=\lim_k\Delta(n_k)$| for some sequence |$(n_k)_k$| satisfying \begin{equation}\label{eq:n_k} \forall k\in{\mathbb N}:\ d_k\mid n_k \;\text{ and }\; s_k\mid n_{k+1}-n_k. \end{equation} (49) Moreover, sequences |$(n_k)_k$| can be defined inductively: For |$n_1$| there are |$s_1/d_1$| choices and, given |$n_1,\dots,n_k$|⁠, there are precisely |$s_{k+1}/{\mathrm{lcm}}(s_k,d_{k+1})$| many choices for |$n_{k+1}$|⁠. (b) |$H_{{\mathrm{int}}(W)}=\{0\}$| if and only if |$s_k=d_k$| for all |$k\in{\mathbb N}$|⁠. □ Remark 5.1. Observe that, in view of (44), \begin{equation*} \frac{s_k}{d_k}\cdot \frac{s_{k+1}}{{\mathrm{lcm}}(s_k,d_{k+1})} = \frac{s_k\, s_{k+1}\gcd(s_k,d_{k+1})}{d_k\,s_k\,d_{k+1}} = \frac{s_k\, s_{k+1}\,d_k}{d_k\,s_k\,d_{k+1}} = \frac{s_{k+1}}{d_{k+1}}, \end{equation*} so that \begin{gather*} \frac{s_k}{d_k}\mid \frac{s_{k+1}}{d_{k+1}}\\ \frac{s_k}{d_k} = \frac{s_1}{d_1}\cdot\prod_{j=1}^{k-1}\frac{s_{j+1}}{{\mathrm{lcm}}(s_j,d_{j+1})}. \end{gather*} □ Proof of Proposition 5.2. (a) For each |$S_k$| denote by |$W_k:=\bigcup_{n\in{\mathcal F}_{{\mathcal A}_{S_k}}}U_{S_k}(\Delta(n))$|⁠. Then |${\mathrm{int}}(W)$| is the increasing union of the sets |$W_k$|⁠, see Lemma 3.1, and |$U_{S_k}(\Delta(n))\subseteq W_k$| if and only if |$U_{S_k}(\Delta(n))\subseteq{\mathrm{int}}(W)$|⁠. Let |$h=\lim_k\Delta(n_k)$|⁠, where |$n_k$| stands for |$n_{S_k}$|⁠, which was defined in Lemma 3.1b. Then \begin{equation} \forall k\in{\mathbb N}:\ s_k\mid n_{k+1}-n_k, \end{equation} (50) and |$h\in H_{{\mathrm{int}}(W)}$|⁠, if and only if \begin{equation}\label{eq:W_k-Delta_n_k} \forall k\in{\mathbb N}:\ {\mathcal F}_{{\mathcal A}_{S_k}}+n_k={\mathcal F}_{{\mathcal A}_{S_k}}. \end{equation} (51) Indeed, let |$k\in{\mathbb N}$|⁠, |$m\in{\mathcal F}_{{\mathcal A}_{S_k}}$|⁠, and let |$g=(g_b)_{b\in{\mathcal B}}$| be any element from |$U_{S_k}(\Delta(m))\subseteq {\mathrm{int}}(W)$|⁠. Then |$g_b=m\mod b$| for all |$b\in S_k$|⁠. Assume now that |$h\in H_{{\mathrm{int}}(W)}$|⁠. Then |$g+h\in{\mathrm{int}}(W)$| and |$(g+h)_b=m+n_k\mod b$| for all |$b\in S_k$|⁠, so that |$g+h\in U_{S_k}(\Delta(m+n_k))$|⁠. Hence |$U_{S_k}(\Delta(m))+h\subseteq U_{S_k}(\Delta(m+n_k))=U_{S_k}(\Delta(m))+\Delta(n_k)$|⁠. In particular, |$U_{S_k}(\Delta(m))$| and |$U_{S_k}(\Delta(m+n_k))$| have identical Haar measure, and so do |$U_{S_k}(\Delta(m))+h$| and |$U_{S_k}(\Delta(m+n_k))$|⁠. As both are open sets and one is contained in the other, they must coincide. Hence |$U_{S_k}(\Delta(m+n_k))=U_{S_k}(\Delta(m))+h\subseteq{\mathrm{int}}(W)+h={\mathrm{int}}(W)$|⁠, so that |$m+n_k\in{\mathcal F}_{{\mathcal A}_{S_k}}$|⁠. This proves that |${\mathcal F}_{{\mathcal A}_{S_k}}+n_k\subseteq{\mathcal F}_{{\mathcal A}_{S_k}}$|⁠. As |${\mathcal A}_{S_k}$| is a finite set, this implies |${\mathcal F}_{{\mathcal A}_{S_k}}+n_k={\mathcal F}_{{\mathcal A}_{S_k}}$|⁠. Conversely, assume that (51) holds, and let |$U_{S_k}(\Delta(m))\subseteq{\mathrm{int}}(W)$|⁠. Recall that this implies |$U_{S_k}(\Delta(m))\subseteq W_k$|⁠, that is |$m\in{\mathcal F}_{{\mathcal A}_{S_k}}$|⁠. Hence, by assumption, also |$m+n_k\in{\mathcal F}_{{\mathcal A}_{S_k}}$|⁠, so that |$U_{S_k}(\Delta(m+n_k))\subseteq W_k\subseteq{\mathrm{int}}(W)$|⁠. Let |$g\in U_{S_k}(\Delta(m))$|⁠. Then |$g_b=m\mod b$| for all |$b\in S_k$|⁠, so that |$(g+h)_b=m+n_k\mod b$| for all |$b\in S_k$|⁠, that is |$g+h\in U_{S_k}(\Delta(m+n_k))$|⁠. Hence |$U_{S_k}(\Delta(m))+h\subseteq U_{S_k}(\Delta(m+n_k))\subseteq{\mathrm{int}}(W)$|⁠. As this argument applies to all |$k$| and all |$U_{S_k}(\Delta(m))\subseteq{\mathrm{int}}(W)$|⁠, it proves that |${\mathrm{int}}(W)+h\subseteq{\mathrm{int}}(W)$|⁠. The same Haar measure argument as before, applied to the open set |${\mathrm{int}}(W)$|⁠, shows that |${\mathrm{int}}(W)+h={\mathrm{int}}(W)$|⁠, that is |$h\in H_{{\mathrm{int}}(W)}$|⁠. Condition (51) is equivalent to \begin{equation} \forall k\in{\mathbb N}:\ c_k={\mathrm{lcm}}({\mathcal A}^{prim}_{S_k})\mid n_k. \end{equation} (52) Invoking Lemma 5.3, we conclude \begin{equation} h\in H_{{\mathrm{int}}(W)} \quad\Leftrightarrow\quad \forall k\in{\mathbb N}:\ d_{k}\mid n_{k}\;\text{ and }\; s_k\mid n_{k+1}-n_k. \end{equation} (53) This proves the claimed equivalence. Now we describe all sequences |$(n_k)_{k\in{\mathbb N}}$| which satisfy (49) and |$n_k\in\{0,\dots,s_k-1\}$| for all |$k$|⁠. Denote |$q_k:=s_k/d_k$|⁠. |$n_1$|⁠: Let |$n_1=m_1d_1$| for any |$m_1\in\{0,\dots,q_1-1\}$|⁠. |$n_2$|⁠: |$n_2$| must be chosen such that |$n_2=0\mod d_2$| and |$n_2=n_1\mod s_1$|⁠. As |$\gcd(s_1,d_2)=d_1\mid n_1$| in view of (44), the CRT guarantees the existence of at least one solution |$n_2$|⁠, and if |$n_2$| is one particular solution, then the set of all solutions is precisely |$n_2+{\mathrm{lcm}}(s_1,d_2)\cdot {\mathbb Z}$|⁠. As |$n_2$| is to be chosen in |$\{0,\dots,s_2-1\}$|⁠, there are exactly |$s_2/{\mathrm{lcm}}(s_1,d_2)$| possible choices for |$n_2$|⁠. |$\vdots$| |$n_{k+1}$|⁠: |$n_{k+1}$| must be chosen such that |$n_{k+1}=0\mod d_{k+1}$| and |$n_{k+1}=n_k\mod s_k$|⁠. As |$\gcd(s_k,d_{k+1})=d_k\mid n_k$| in view of (44), the CRT guarantees the existence of at least one solution |$n_{k+1}$|⁠, and if |$n_{k+1}$| is one particular solution, then the set of all solutions is precisely |$n_{k+1}+{\mathrm{lcm}}(s_k,d_{k+1})\cdot {\mathbb Z}$|⁠. As |$n_{k+1}$| is to be chosen in |$\{0,\dots,s_{k+1}-1\}$|⁠, there are exactly |$s_{k+1}/{\mathrm{lcm}}(s_k,d_{k+1})$| possible choices for |$n_{k+1}$|⁠. (b) |$H_{{\mathrm{int}}(W)}=\{0\}$||$\Leftrightarrow$| there is unique choice of the numbers |$n_k$| described in a) |$\Leftrightarrow$||$s_1/d_1=1$| and |$s_{k+1}/{\mathrm{lcm}}(s_k,d_{k+1})=1$| for any |$k$||$\Leftrightarrow$||$d_{k}=s_{k}$| for any |$k$|⁠, the last equivalence by Remark 5.1. ■ 5.2 Proof of Theorem D Remark 5.2. If |$(S_k)_k$| is a filtration of |${\mathcal B}$| by finite sets and if |$h=(h_b)_{b\in{\mathcal B}}\in H$|⁠, then we write |$ \lim_k\Delta(n_{S_k})=h$|⁠, whenever |$n_{S_k}\in{\mathbb Z}$| are numbers such that for every |$k\in {\mathbb N}$|⁠: $$ h_b= n_{S_k}\mod b\;{\rm {for\,\,all}}\;b\in S_k. $$ Let us denote |$s_k={\mathrm{lcm}}(S_k)$|⁠. There is an inverse system of groups \begin{equation*} \ldots {\mathbb Z}/s_{k+1}{\mathbb Z}\rightarrow {\mathbb Z}/s_{k}{\mathbb Z} \rightarrow \ldots\rightarrow {\mathbb Z}/s_1{\mathbb Z}. \end{equation*} The homomorphisms are the canonical projections. Observe that |$s_k|n_{S_{k+1}}-n_{S_k}$| for any |$k$| and the sequence |$(n_{S_k}+s_k{\mathbb Z})_k$| is an element of the inverse limit |$\lim\limits_{\leftarrow}{\mathbb Z}/{s_k}{\mathbb Z}$|⁠. In this way we obtain an isomorphism of topological groups \begin{equation}\label{sigma} \sigma:\lim\limits_{\leftarrow}{\mathbb Z}/{s_k}{\mathbb Z}\cong H \end{equation} (54) given by |$(n_{S_k}+s_k{\mathbb Z})_k\mapsto \lim_k\Delta(n_{S_k})$|⁠. Compare Remark 2.32 [4]. In particular, the inverse limit does not depend on the filtration |$(S_k)_k$|⁠. (This last statement follows from a general property of inverse limits: the inverse limits of cofinal inverse systems are isomorphic, [9, Chapter II, Section 12].) □ Proof of Proposition 1.3 Let |$\beta_k:{\mathbb Z}/s_k{\mathbb Z}\rightarrow {\mathbb Z}/d_k{\mathbb Z}$| be the map given by |$n+s_{k}{\mathbb Z}\mapsto n+d_{k}{\mathbb Z}$|⁠, let |$M_k$| be the kernel of |$\beta_k$| and let |$\alpha_k:M_k\rightarrow {\mathbb Z}/s_k{\mathbb Z}$| be the canonical embedding. There is a commutative diagram of abelian groups where |$f_k(n+s_k{\mathbb Z})=n+s_{k-1}{\mathbb Z}$|⁠, |$f'_k$| is the restriction of |$f_k$| to |$M_k$| and |$f''_k(n+d_k{\mathbb Z})=n+d_{k-1}{\mathbb Z}$|⁠. The columns of the diagram are exact sequences of groups, in other words, the diagram can be interpreted as an exact sequence of inverse systems of abelian groups. Since inverse limit is a left exact functor, see [9, Chapter II, Theorem 12.3], we obtain an exact sequence \begin{equation}\label{alg_2} 0\rightarrow \lim\limits_{\leftarrow}M_k{\smash{\mathop{\longrightarrow}\limits^{{\alpha}}_{{}}}} \lim\limits_{\leftarrow}{\mathbb Z}/s_k{\mathbb Z}{\smash{\mathop{\longrightarrow}\limits^{{\beta}}_{{}}}} \lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}. \end{equation} (55) The condition (44) yields that the homomorphism |$\gamma$| in (55) is surjective, thus we have an exact sequence \begin{equation}\label{alg_3} 0\rightarrow \lim\limits_{\leftarrow}M_k{\smash{\mathop{\longrightarrow}\limits^{{\alpha}}_{{}}}} \lim\limits_{\leftarrow}{\mathbb Z}/{s_k}{\mathbb Z}{\smash{\mathop{\longrightarrow}\limits^{{\beta}}_{{}}}} \lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}\rightarrow 0 \end{equation} (56) Indeed, let |$(n_k+d_k{\mathbb Z})_k\in \lim\limits_{\leftarrow}{\mathbb Z}/{d_k}{\mathbb Z}$|⁠. By induction we construct the numbers |$m_1,m_2,\ldots$| such that |$d_k|m_k-n_k$| and |$s_k|m_{k+1}-m_k$|⁠, for any |$k$|⁠. Then |$\beta((m_k+s_k{\mathbb Z})_k)=(n_k+d_k{\mathbb Z})_k$|⁠. We set |$m_1=n_1$|⁠. Assume that |$m_1,\ldots m_k$| have been defined. Since |$d_k\mid n_{k+1}-n_k$|⁠, |$d_k\mid m_{k}-n_k$| and |$\gcd(d_{k+1},s_k)=d_k$|⁠, there exist integers |$x,y$| such that |$xd_{k+1}+ys_k=m_k-n_{k+1}$|⁠. We set |$m_{k+1}=m_k-ys_k$|⁠. There are group isomorphisms |$g_k:{\mathbb Z}/{\frac{s_{k}}{d_{k}}}{\mathbb Z} \rightarrow M_k$| given by |$g_k(n+\frac{s_{k}}{d_{k}}{\mathbb Z})=d_kn+s_k{\mathbb Z}$| and making the following diagram commutative (the arrows in the upper row represent the canonical projections). It follows that there is an isomorphism \begin{equation}\label{alg_4} \lim\limits_{\leftarrow}M_k\cong \lim\limits_{\leftarrow}{\mathbb Z}/{\frac{s_{k}}{d_{k}}}{\mathbb Z}. \end{equation} (57) By Proposition 5.2 a) it follows that |$\lim\limits_{\leftarrow}M_k$| is isomorphic to |$H_{{\mathrm{int}}(W)}$|⁠. There is an isomorphism given by |$\sigma\alpha$|⁠, where |$\sigma$| is the isomorphism defined in Remark 5.2. Now a), b) and c) follow from (56), (57) and Remark 5.2. In order to prove d) it is enough to note that |$s_k=d_k$| if and only if |$s_k\mid c_{k+j}$| for some |$j\ge 0$|⁠. ■ Proof of Theorem D This is an immediate corollary to Proposition 1.3. ■ 5.3 Examples Remark 5.3. Given a prime number |$p$| and |$m\in{\mathbb Z}$| we denote by |$v_p(m)$| be the |$p$|-valuation of |$m$|⁠, that is, if |$m\neq 0$| then |$v_p(m)$| is the maximal integer such that |$p^{v_p(m)}\mid m$| and |$v_p(0)=+\infty$|⁠. Assume that |$t=(t_k)$| is a sequence of natural numbers such that |$t_k\mid t_{k+1}$| for any |$k$|⁠. Set |$v_p(t)=\sup_kv_p(t_k)$|⁠. The sequence |$t$| yields an inverse system of abelian groups $$ \ldots\rightarrow Z/t_{k+1}{\mathbb Z}\rightarrow Z/t_{k}{\mathbb Z} \rightarrow\ldots \rightarrow Z/t_{1}{\mathbb Z}, $$ where the arrows represent the canonical projections |$n+t_{k+1}{\mathbb Z}\mapsto n+t_{k}{\mathbb Z}$|⁠. The inverse limit |$\lim\limits_{\leftarrow}{\mathbb Z}/t_{k}{\mathbb Z}$| of this system is isomorphic to the group $$ \prod\limits_{p\in{\mathcal P}}G_p, $$ where |$G_p={\mathbb Z}/p^{v_p(t)}{\mathbb Z}$| if |$v_p(t)<+\infty$| and |$G_p=\widehat{{\mathbb Z}}_p$| (the group of |$p$|-adic numbers) otherwise, that is when |$\lim_kv_p(t_k)=+\infty$|⁠. □ Recall from (6) that \begin{equation} {{\mathcal A}_\infty}=\{c\in{\mathbb N}: \forall_{S\subset{\mathcal B}}\ \exists_{S': S\subseteq S'}: c\in{\mathcal A}_{S'}\setminus S'\}. \end{equation} (58) Our first exaxmple has a finite, non-trivial maximal equicontinuous factor and a finite set |${{\mathcal A}_\infty}$|⁠. Example 5.1. |${\mathcal B}=\{36\}\cup \{2p_1,2p_2,\ldots\}\cup \{3q_1,3q_2,\ldots\}$|⁠, where |$p_1,q_1,p_2,q_2,\ldots$| are pairwise different primes. Let |$S_k=\{36,2p_1,\dots,2p_k,3q_1,\dots,3q_k\}$|⁠. Then \begin{equation*} s_k=36p_1\cdots p_kq_1\cdots q_k,\; {{\mathcal A}_{S_k}=\{2,3\}\cup S_k},\;c_k=d_k=6, \end{equation*} so that, \begin{equation*} \frac{s_k}{d_k}=6p_1\cdots p_kq_1\cdots q_k. \end{equation*} In particular, the maximal equicontinuous factor of |$X_\eta$| is the translation by |$1$| on |${\mathbb Z}/6{\mathbb Z}$|⁠. Moreover, |${{\mathcal A}_\infty}=\{2,3\}$|⁠, so that |$\emptyset\neq\overline{{\mathrm{int}}(W)}\neq W$| by Theorems B and C. □ Our next example has an infinite maximal equicontinuous factor different from |$H$| and an infinite set |${{\mathcal A}_\infty}$|⁠. Example 5.2. Let |$p_1,q_1,p_2,q_2,\ldots$| be pairwise different primes. Let $$ {\mathcal B}={\mathcal B}_1\cup{\mathcal B}_2\cup{\mathcal B}_3\ldots, $$ where $$ \begin{array}{l} {\mathcal B}_1=\{p_1q_1\}\\ {\mathcal B}_2=\{p_1p_2^2, p_1q_2^2, q_1q_2^2\}\\ {\mathcal B}_3=\{p_1p_2p_3^2,p_1p_2q_3^2,p_1q_2q_3^2, q_1q_3^2\}\\ {\mathcal B}_4=\{p_1p_2p_3p_4^2,p_1p_2p_3q_4^2, p_1p_2q_3q_4^2, p_1q_2q_4^2, q_1q_4^2\}\\ {\mathcal B}_5=\{p_1p_2p_3p_4p_5^2,p_1p_2p_3p_4q_5^2,p_1p_2p_3q_4q_5^2,p_1p_2q_3q_5^2, p_1q_2q_5^2, q_1q_5^2\}\\ \ldots \end{array}$$ That is, \begin{align*} {\mathcal B}_{k+1} & =\{p_1\ldots p_k p_{k+1}^2,\;p_1\ldots p_k q_{k+1}^2,\;p_1\ldots p_{k-1} q_k q_{k+1}^2\}\\ &\quad{}\cup\left\{\frac{b q_{k+1}^2}{q_k^2}:b\in{\mathcal B}_k\setminus\{p_1\ldots p_{k-1}p_{k}^2,p_1\ldots p_{k-1 }q_{k}^2\}\right\} \end{align*} for |$k\ge 2$|⁠. Let |$S_k={\mathcal B}_1\cup\ldots\cup{\mathcal B}_k$|⁠. Then |$s_k={\mathrm{lcm}}(S_k)=p_1p_2^2\ldots p_k^2q_1q_2^2\ldots q_k^2$| and $$ {\mathcal A}_{S_k}=S_k\cup\{p_1\ldots p_k,\;p_1\ldots p_{k-1}q_k,\; p_1\ldots p_{k-2}q_{k-1},\;\ldots, p_1q_2,q_1\}, $$ so that, \begin{equation*} {\mathcal A}^{prim}_{S_k} = \{p_1\ldots p_k,\;p_1\ldots p_{k-1}q_k,\; p_1\ldots p_{k-2}q_{k-1},\;\ldots, p_1q_2,q_1\}. \end{equation*} Hence, \begin{equation*} c_k=p_1\cdots p_k q_1\cdots q_k\quad\text{and}\quad d_k=\gcd(s_k,c_{k+j})=c_k, \end{equation*} so that, \begin{equation*} \frac{s_k}{d_k}=p_2\cdots p_kq_2\cdots q_k. \end{equation*} Hence |$H_{{\mathrm{int}}(W)}\cong\prod_{i=2}^{+\infty}{\mathbb Z}/p_iq_i{\mathbb Z}$| and |$H/H_{{\mathrm{int}}(W)}\cong\prod_{i=1}^{+\infty}{\mathbb Z}/p_iq_i{\mathbb Z}$| are infinite compact groups. 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Google Scholar Crossref Search ADS [16] Sarnak P. “Three lectures on Möbius function, randomness and dynamics.” publications.ias.edu/sarnak/ . © The Author(s) 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Cadoret,, Anna;Tamagawa,, Akio

doi: 10.1093/imrn/rnx193pmid: N/A

Abstract Let |$X$| be a connected scheme, smooth and separated over an algebraically closed field |$k$| and let |$\rho_{\ell}:\pi_{1}(X)\rightarrow \textrm{GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$\ell\in L$| be a family (indexed by an infinite set |$L$| of primes) of continuous |$\mathbb{F}_{\ell}$|-linear representations of the étale fundamental group of |$X$| of bounded degree |$r_{\ell}\leq r$|⁠. The most important examples of such families are those arising from the étale cohomology with |${\mathbb{F}}_\ell$|-coefficients of the geometric generic fiber of a smooth proper scheme over |$X$|⁠. The main result of this article asserts that, under a mild finiteness assumption, the image |$G_{\ell}$| of |$\rho_{\ell}:\pi_{1}(X)\rightarrow {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$\ell\in L$| is “almost algebraic” for |$\ell\gg 0$|⁠. This is the analog for |${\mathbb{F}}_\ell$|-coefficients of Grothendieck’s unipotency theorem—a crucial step in the proof of Deligne’s semisimplicity theorem in Weil II. Just as for |${\mathbb{Q}}_\ell$|-coefficients, our result is a crucial step to establish the analog of Deligne’s semisimplicity theorem for |${\mathbb{F}}_\ell$|-coefficients (proved by Chun Yin Hui and the authors in a subsequent article). Our result also has a wide range of other applications—in particular to the variation of invariants in one-dimensional families of varieties and the existence of closed Galois-generic points for motivic representations. We give a first simple example of such applications in the final section of this article. 1 Introduction The étale fundamental group of a scheme [22] is one of the most elaborated tool in arithmetic geometry. It enables one to translate intricate geometric problems into representation-theoretic ones, easier to handle. As a result, representations of the étale fundamental group—especially the ones arising from étale cohomology—are ubiquitous. For instance, those with |${\mathbb{Q}}_\ell$|-coefficients play a crucial part in the proof of the Weil conjectures [17, 18] and are at the heart of the Langlands program for function fields [29, 30] or of the Grothendieck–Serre–Tate conjectures [37]. They also are the natural tool to study the variation of motivic invariants in families [5, 8, 12, 13]. One remarkable consequence of Deligne’s weight theory is the semisimplicity of the Zariski-closure of the image of the geometric monodromy acting on |$\ell$|-adic cohomology [18, Corollary 3.4.13]—a significant step towards the Grothendieck–Serre semisimplicity conjecture. More precisely, Deligne’s semisimplicity theorem is the combination of two independent results: (1) the radical of the image is unipotent (namely, Grothendieck’s unipotency theorem—[18, Theorem 1.3.8, Corollary 1.3.9]), which is deduced from class field theory [18, Theorem 1.3.1, Proposition 1.3.4] by a purely group-theoretic argument; (2) the representation is semisimple (equivalently, the radical of the image is a torus) [18, Corollary 3.4.13], which is deduced from Deligne’s weight theory. Representations with |${\mathbb{F}}_\ell$|-coefficients are less understood. They contain additional information but this information is more difficult to capture. One reason is the lack of a “well-behaved Zariski-closure” for finite subgroups of |${\rm GL}_r({\mathbb{F}}_\ell)$|⁠. However, for finite subgroups |$G\subset {{\rm GL}}_r({\mathbb{F}}_\ell)$| which are generated by their |$\ell$|-Sylow subgroups, Nori constructed an algebraic envelope, which is a right substitute for the “well-behaved Zariski-closure” [32]. More recently, Larsen and Pink developed a different approach leading to a variant of the notion of algebraic envelope, also suitable for finite subgroups |$G \subset {{\rm GL}}_r(\overline{{\mathbb{F}}}_\ell)$| [31]. The main result of this article—Theorem 1.1 below—is the analog for |${\mathbb{F}}_\ell$|-coefficients of Grothendieck’s unipotency theorem (1) in the proof of Deligne’s semisimplicity theorem. The proof of the semisimplicity theorem for |${\mathbb{F}}_\ell$|-coefficients is completed in subsequent papers: in [15], we establish that (1’) the images are “almost” perfect and, in [9], we prove that (2) the representations are semisimple. Contrary to the proof for |${\mathbb{Q}}_\ell$|-coefficients, where (2) is independent from (1), for |${\mathbb{F}}_\ell$|-coefficients, the order in which we prove (1), (1’), and (2) is crucial. Namely, (2) uses both (1’) and (1), and (1’) uses (1). The reason why, for |${\mathbb{F}}_\ell$|-coefficients, (1) has to come first is that it establishes both the algebraicity of the image of the geometric monodromy—which is “automatic” for |${\mathbb{Q}}_\ell$|-coefficients and the fact that it is generated by its unipotent subgroups. A first kind of applications of (1) and (1’) for |${\mathbb{F}}_\ell$|-coefficients is to the variation of invariants in one-dimensional families of algebraic varieties, where the key-point is to obtain lower bounds for the genus [15] and gonality [14] of abstract modular curves with level-|$\ell$| structures. These are the analogs for |${\mathbb{F}}_\ell$|-coefficients of the results of [12, 13]. Another kind of application of Theorem 1.1 is the existence of closed Galois-generic points [6, 7, 10]. We give a first simple example of such applications in Section 4. Let us now state our main result and describe more precisely the content of this article. Let |$k$| be a field of characteristic |$p\geq 0$| and let |$X$| be a scheme geometrically connected, smooth and separated over |$k$|⁠. Let |$L$| be an infinite set of primes and let $$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$$ be a family of continuous |$\mathbb{F}_{\ell}$|-linear representations of the étale fundamental group of |$X$| with |$r_{\ell}\leq r$| bounded as |$\ell$| varies. Typical examples of such families are those arising from the étale cohomology with coefficients in |$\mathbb{F}_{\ell}$| of the geometric generic fiber of a smooth proper morphism |$Y\rightarrow X$|⁠. Assume that |$k$| is algebraically closed. Given a subgroup |$G\subset {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, write |$G[\ell]\subset G$| for the set of all |$g\in G$| such that |$g^{\ell}=1$| (for |$\ell\geq r$|⁠, there is no element of order |$\ell^{2}$| in GL|$_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠) and |$G^{+}\subset G$| for the (normal) subgroup generated by |$G[\ell]$|⁠. Then the following main result of this article asserts that under mild finiteness assumption—Condition (F) in Section 3.1—the images of the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$| are generated by their |$\ell$|-Sylow and “almost algebraic” for |$\ell\gg 0$|⁠. Theorem 1.1. Assume that Condition (F) holds. Then there exists an open subgroup |$\Pi\subset \pi_{1}(X)$| such that $$\rho_{\ell}(\Pi)=\rho_{\ell}(\Pi)^{+}$$ for |$\ell\gg 0$| (depending on |$\Pi$|⁠). Furthermore, for any such |$\Pi$| and every open subgroup |$\Pi'\subset \Pi$|⁠, one also has |$\rho_{\ell}(\Pi')=\rho_{\ell}(\Pi')^{+}(=\rho_{\ell}(\Pi))$| for |$\ell\gg 0$| (depending on |$\Pi'$|⁠). □ The terminology “almost algebraic” comes from the following. Write |$\mathcal{G}\hookrightarrow {\rm GL}_{r_{\ell}/\mathbb{F}_{\ell}}$| for the algebraic envelope of |$G$| that is the algebraic subgroup generated by the |$1$|-parameter subgroups \begin{align*} e_{g}:\mathbb{A}^{1}_{\mathbb{F}_{\ell}}&\rightarrow{\rm GL}_{r_{\ell}/\mathbb{F}_{\ell}}\\ t&\mapsto{\rm exp}(t{\rm log}(g)) ,\; g\in G[\ell]. \end{align*} Then, we have Theorem 1.2. [32, Theorem B and Remark 3.6] For |$\ell\gg 0$| (depending only on |$r$|⁠) and for every subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠, one has $$ G\supset G^{+}=\mathcal{G}(\mathbb{F}_{\ell})^{+}\subset \mathcal{G}(\mathbb{F}_{\ell}) $$ and |$\mathcal{G}(\mathbb{F}_{\ell})/\mathcal{G}(\mathbb{F}_{\ell})^{+}$| is an abelian group of order |$\leq 2^{r-1}$|⁠. □ The proof of Theorem 1.1 reduces easily to the case where the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$| are semisimple and then, decomposes into two steps—Claims 1 and 2 in Section 3.2. Condition (F) may seem technical at first glance but just says that we may always assume that our base scheme is a curve and that the representations are almost tame (which, again, is automatic in the setting of |${\mathbb{Q}}_\ell$|-coefficients). It is there to make the arguments in the proof of Claim 2 work. Claim 2 has to be thought of as the analog of class field theory in Grothendieck’s unipotency theorem (note that class field theory is only available for étale fundamental group of curves!) while Condition (F) is the substitute of the specialization step in Deligne’s argument (See [18, (1.11.4) and Variante after Proposition 1.3.4]{DeligneWCII}). The fact that it is satisfied by representations arising from étale cohomology is nontrivial and is established in Proposition 3.2. Eventually, Claim 1 is the analog of the group-theoretic argument in Grothendieck’s unipotency theorem and just as it, is purely group-theoretic. More precisely, it is based on a refinement (Theorem 2.2) of a theorem of Nori [32, Theorem C] about the structure of subgroups of GL|$_{r}(\mathbb{F}_{\ell})$|⁠. We should mention that the full strength of Theorem 2.2 is not used in the proof of Claim 1. Indeed, for semi-simple representations, Theorem 2.2 is probably well-known to specialists and, as pointed out by a referee, could be reconstructed from ingredients in the existing literature (such as those in [28]; see also Remark 2.10). However, to keep the exposition elementary and self-contained and also because we feel Theorem 2.2 is interesting in itself and might have further applications, we give a full detailed proof, based only on Nori theory. The article is organized as follows. Section 2 is devoted to the statements and proofs of the group-theoretical preliminaries required for Claim 1. Section 3 is devoted to the structure of the geometric images of the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$|⁠. In Ssection 3.1, we formulate Condition (F) and show that it is satisfied by families of |$\mathbb{F}_{\ell}$|-linear representations arising from the étale cohomology with coefficients in |$\mathbb{F}_{\ell}$| of the geometric generic fiber of a smooth proper morphism |$Y\rightarrow X$|⁠. The proof of Theorem 1.1 (including the proofs of Claims 1 and 2) is carried out in Ssection 3.2. We also state there two corollaries (Corollaries 3.3 and 3.5) which refine Theorem 1.1 when the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$| are semisimple. Corollary 3.3 enables one to extend, for instance, the main result of [19] (for families of representations arising from the |$\ell$|-torsion of abelian schemes) to families of representations arising from étale cohomology in characteristic |$0$|⁠. In Section 4, we give another application of Theorem 1.1, to the problem of almost |$\ell$|-independence (in the sense of Serre—[36]) for families of |$\ell$|-adic representations. This is strengthened and developed in subsequent works of the first author [6, 7, 10]. 2 Group-Theoretical Preliminaries 2.1 Statements An old theorem of Jordan [25] asserts that there exists an integer |$\delta(r)\geq 1$| such that every finite subgroup of GL|$_{r}(\mathbb{C})$| has a normal abelian subgroup of index |$\leq \delta(r)$|⁠. Nori [32] extended this result to subgroups of GL|$_{r}(\mathbb{F}_{\ell})$| as follows. Theorem 2.1. [32, Theorem C] For every integer |$r\geq 1$| there exists an integer |$\delta(r)\geq 1$| such that for every prime |$\ell$| and for every subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| there exists a subgroup |$T\subset G$| such that - |$T$| is abelian of prime-to-|$\ell$| order; - |$G^{+}T$| is normal in |$G$|⁠; - |$[G:G^{+}T]\leq \delta(r)$|⁠. □ In particular, there exist normal subgroups |$G_{2}\subset G_{1}\subset G$| satisfying the following properties - |$[G:G_{1}]\leq \delta(r)$|⁠; - |$G_{1}/G_{2}$| is abelian of prime-to-|$\ell$| order; - |$G_{2}=G_{2}^{+}(=G^{+})$|⁠. We improve Theorem 2.1 by showing that |$T$| can be chosen in such a way that it centralizes |$G^{+}$| provided |$G^{+}$| acts semi-simply on |$\mathbb{F}_{\ell}^{\oplus r}=:H_{\ell}$|⁠. This will be a consequence of the following slightly more general result. Theorem 2.2. Fix an integer |$d \geq 1$|⁠. Then, for every integer |$r\geq 1$| there exists an integer |$\delta(d,r)\geq 1$| such that for every prime |$\ell$| and for every subgroup |$ G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| and subgroups |$\Delta ,T \subset G $| satisfying the following properties (1) |$\Delta=\Delta^{+}$| and it acts semisimply on |$H_{\ell}$|⁠; (2) |$\Delta $| is normal in |$G $|⁠; (3) |$T$| is abelian (resp. abelian of prime-to-|$\ell$| order); (4) |$\Delta T $| is normal in |$G$|⁠; (5) |$[G:\Delta T]\leq d$|⁠, there exists a subgroup |$\Theta \subset G $| satisfying the following properties - |$\Theta $| is abelian (resp. abelian of prime-to-|$\ell$| order); - |$\Theta $| centralizes |$\Delta$|⁠; - |$\Theta$| (hence |$\Delta \Theta $|⁠) is normal in |$G$|⁠; - |$[G :\Delta\Theta]\leq \delta(d,r)$|⁠. □ In particular, combining Theorems 2.2 and 2.1 we obtain the following general structure result for subgroups of |${\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠. Corollary 2.3. For every integer |$r\geq 1$| there exists an (explicit) integer |$\delta(r)\geq 1$| such that for every prime |$\ell$| and for every subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| such that |$G^{+}$| acts semisimply on |$H_{\ell}$|⁠, there exists a subgroup |$T\subset G$| satisfying the following properties - |$T$| is abelian of prime-to-|$\ell$| order; - |$T$| centralizes |$G^{+}$|⁠; - |$T$| (hence |$G^{+}T$|⁠) is normal in |$G$|⁠; - |$[G :G^{+}T]\leq \delta(r)$|⁠. □ From which, in turn, one recovers a slight variant of the filtration of [31, Theorem 0.2] for |$k=\mathbb{F}_{\ell}$|⁠. Corollary 2.4. For every integer |$r\geq 1$| there exists an (explicit) integer |$\delta(r)\geq 1$| such that for every prime |$\ell$| and for every subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| there exist normal subgroups |$G_{3}\subset G_{2}\subset G_{1}\subset G$| satisfying the following properties - |$[G:G_{1}]\leq \delta(r)$|⁠; - |$G_{1}/G_{2}\twoheadleftarrow\mathcal{G}(\mathbb{F}_{\ell})^{+}(=[\mathcal{G}(\mathbb{F}_{\ell}),\mathcal{G}(\mathbb{F}_{\ell})])$| for some semisimple algebraic subgroup |$\mathcal{G}\hookrightarrow {\rm GL}_{r/\mathbb{F}_{\ell}}$|⁠; - |$G_{2}/G_{3}$| is abelian of prime-to-|$\ell$| order; - |$G_{3}$| is an |$\ell$|-group. □ Proof. First, let us observe that it is enough to prove Corollary 2.4 for |$\ell\gg 0$| (depending only on |$r$|⁠). Indeed, assume there exists a prime |$\ell(r)$| such that Corollary 2.4 holds for every prime |$\ell\geq \ell(r)$| with a given constant |$\delta^{0}(r)$|⁠. Then, up to replacing |$\delta^{0}(r)$| with |$\delta(r):=\delta^{0}(r)|{\rm GL}_{r}(\mathbb{F}_{\ell(r)})|$|⁠, the statement of Corollary 2.4 still holds for primes |$\ell<\ell(r)$| (with |$G_{3}=G_{2}=G_{1}=1$|⁠). So, in the proof of Corollary 2.4, we may consider only primes |$\ell\gg 0$| (depending only on |$r$|⁠). Given a subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠, consider the semi-simplification |$H_{\ell}^{ss}$| of |$H_{\ell}$| as a |$G$|-module and let |$G\twoheadrightarrow G^{ss}$| denote the image of |$G$| acting on |$H_{\ell}^{ss}$|⁠. Let |$G_{3}$| denote the kernel of |$G\twoheadrightarrow G^{ss}$|⁠. Let |$T\subset G^{ss}$| denote any subgroup satisfying the conclusions of Corollary 2.3 (applied to |$G^{ss}\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠) and let |$G_{2}$| and |$G_{1}$| be the inverse images of |$T$| and |$G^{ss,+}T$| by |$G\twoheadrightarrow G^{ss}$| respectively. By construction, |$G_{3}\subset G_{2}\subset G_{1}\subset G$| are normal subgroups. Also, |$G_{3}$| is a unipotent group hence an |$\ell$|-group; |$T\simeq G_{2}/G_{1}$| is abelian of prime-to-|$\ell$| order; |$[G:G_{1}]=[G^{ss}:G^{ss,+}T]\leq \delta(r)$|⁠; |$G^{ss,+}$| surjects onto |$G_{1}/G_{2}$| and, by Theorem 1.2 and Lemma 2.6 below, |$G^{ss,+}=\mathcal{G}(\mathbb{F}_{\ell})^{+}$| for some semi-simple algebraic subgroup |$\mathcal{G}\hookrightarrow {\rm GL}_{r/\mathbb{F}_{\ell}}$| and for |$\ell\gg 0$| (depending on |$r$|⁠). Also, from Theorem 1.2 again, |$[\mathcal{G}(\mathbb{F}_{\ell}),\mathcal{G}(\mathbb{F}_{\ell})]\subset G^{ss,+}$|⁠. But from Lemma 3.4 below, |$G^{ss,+}=[G^{ss,+},G^{ss,+}](\subset [\mathcal{G}(\mathbb{F}_{\ell}),\mathcal{G}(\mathbb{F}_{\ell})])$| for |$\ell\gg 0$| (depending on |$r$|⁠). ■ 2.2 Proof of Theorem 2.2 Before turning to the proof of Theorem 2.2 itself, let us observe that it is enough to prove it for |$\ell\gg 0$| (depending only on |$r$|⁠). Indeed, assume there exists a prime |$\ell(r)$| such that Theorem 2.2 holds for every prime |$\ell\geq \ell(r)$| with a given constant |$\delta^{0}(d,r)$|⁠. Then, up to replacing |$\delta^{0}(d,r)$| with |$\delta(d,r):={\rm max}\lbrace \delta^{0}(d,r),|{\rm GL}_{r}(\mathbb{F}_{\ell(r)})|\rbrace$|⁠, the statement of Theorem 2.2 still holds for primes |$\ell<\ell(r)$| (with |$\Theta=1$|⁠). So, in the proof of Theorem 2.2, we may consider only primes |$\ell\gg 0$| (depending only on |$r$|⁠). Also, it is enough to prove Theorem 2.2 with the seemingly weaker conclusion that |$\Delta\Theta$| is normal in |$G$| (instead of |$\Theta$| being normal in |$G$|⁠). Indeed, then the center of |$ \Delta\Theta$| (resp. the prime-to-|$\ell$| part of the center of |$\Delta\Theta$|⁠) will satisfy the conclusions of Theorem 2.2. 2.2.1 Strategy The basic idea is to show that |$T$| can be “adjusted” in such a way that it centralizes |$\Delta$|⁠. Roughly, let |$c:G\rightarrow \textrm{ Aut}(\Delta)$| denote the morphism induced by conjugation and consider the following diagram, whose lower sequence is exact. Here, |$Z(\Delta)$|⁠, |$\textrm{Aut}(\Delta),$| and |${\rm Out}(\Delta)$| denote the center, group of automorphisms and group of outer automorphisms of |$\Delta$| respectively. Assume that up to replacing |$T$| by a subgroup of index bounded only in terms of |$r$| and still satisfying properties (4), (5) (with |$d$| replaced) of Theorem 2.2 and |$c$| by the |$n$|th power of |$c$| for some |$n>0$| bounded only in terms of |$r$|⁠, one can lift successively |$c$| to |$c_{1}$| and |$c_{2}$|⁠. Then it is not difficult to see that $$\Theta:=\lbrace t^n c_{2}(t)^{-1}\; |\; t\in T\rbrace\subset G$$ will satisfy the conclusion of Theorem 2.2. Lifting |$c_{1}$| to |$c_{2}$| is a purely group-theoretical problem, which can be solved elementarily (see Section 2.2.2.2). Lifting |$c$| to |$c_{1}$| is more delicate. (In this process, we may have to replace |$Z(\Delta)$| by a possibly smaller subgroup |$Z\subset Z(\Delta)$|⁠, but this will not affect the final purely group-theoretical lifting step). It would be straightforward if |$|{\rm Out}(\Delta)|$| were bounded above by a constant depending only on |$r$|⁠. Such a constant does not exist a priori for abstract finite groups |$\Delta$|⁠. But it does for semisimple algebraic subgroups of GL|$_{r/\mathbb{F}_{\ell}}$|⁠. More precisely, one has (see e.g., [16, Proposition 7.1.6, Remark 7.1.7, and Theorem 7.1.9]): Theorem 2.5. Let |$\mathcal{G}$| be a semi-simple algebraic group over a field |$k$|⁠. Then the functor of automorphisms |$\underline{{\rm Aut}}_{k}(\mathcal{G})$| is represented by a smooth algebraic group |${\rm Aut}(\mathcal{G})$| over |$k$|⁠, which fits into a short exact sequence of algebraic groups over |$k$| $$ 1\rightarrow \mathcal{G}/Z(\mathcal{G})\rightarrow {\rm Aut}(\mathcal{G})\rightarrow {\rm Out}(\mathcal{G})\rightarrow 1.$$ Furthermore |${\rm Out}(\mathcal{G})$| is finite, étale over |$k$| of degree |$\leq 6^{{\rm \tiny rank}(\mathcal{G})}$|⁠. □ The idea is thus to replace |$\Delta$| by its algebraic envelope |$\mathcal{D}\hookrightarrow {\rm GL}_{r/\mathbb{F}_{\ell}}$|⁠, which is a semisimple algebraic group as soon as |$\Delta$| acts semisimply on |$H_{\ell}$|⁠. Lemma 2.6. Let |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| be a subgroup. Assume that |$H_{\ell}$| is a semisimple |$G^{+}$|-module. Then the algebraic envelope |$\mathcal{G}\hookrightarrow {\rm GL}_{r/\mathbb{F}_{\ell}}$| of |$G$| is a connected semisimple algebraic group. □ Proof. From [3, Proposition 2.2 and its proof], one already knows that |$\mathcal{G}$| connected. - Reductivity: By definition, |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$| is the algebraic group generated by the set $$Y:=\left\{ {\rm exp}(t{\rm log}(g))\; |\; t\in \overline{\mathbb{F}}_{\ell},\; g\in G[\ell]\right\}$$ in |${\rm GL}_{r/\overline{\mathbb{F}}_{\ell}}$|⁠. Observe first that |$H_{\ell\overline{\mathbb{F}}_{\ell}}:=H_{\ell}\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}}_{\ell}$| is a semisimple |$G^{+}$|-module; this follows from the facts that |$\mathbb{F}_{\ell}$| is a perfect field and that |$H_{\ell}$| is a semisimple |$G^{+}$|-module by assumption. Now any |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$|-stable subspace |$V\subset H_{\ell\overline{\mathbb{F}}_{\ell}}$| is a |$G^{+}$|-submodule hence there exists a |$G^{+}$|-stable submodule |$V'\subset H_{\ell\overline{\mathbb{F}}_{\ell}}$| such that |$ H_{\ell\overline{\mathbb{F}}_{\ell}}=V\oplus V'$|⁠. But |$V'$| is then |$Y$|-stable by construction hence |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$|-stable. This shows that |$H_{\ell}$| is a faithful semisimple representation of |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$| hence, in particular, that |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$| is reductive. - Semisimplicity: By construction |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$| is generated by unipotent elements hence has no quotient isomorphic to a torus. This shows that |$\mathcal{G}_{/\overline{\mathbb{F}}_{\ell}}$| is semisimple. ■ 2.2.2 Proof of Theorem 2.2 By definition of the algebraic envelope |$\mathcal{D}$|⁠, the action of |$G$| by conjugation on |$\Delta$| induces a morphism |$c:G\rightarrow {\rm Aut}(\mathcal{D})(\mathbb{F}_{\ell})$|⁠. Consider the following diagram, whose lower sequence is exact. We are going to prove that up to replacing |$T$| by a subgroup of index bounded only in terms of |$r$| and still satisfying properties (4), (5) (with |$d$| replaced) of Theorem 2.2 and |$c$| by the |$n$|th power of |$c$| for some |$n>0$| bounded in terms of |$r$|⁠, we can lift successively |$c$| to |$c_{1,1}$|⁠, |$c_{1,2}$|⁠, and |$c_{1,3}$| (and it is the group |$Z:=\Delta\cap Z(\mathcal{D})(\mathbb{F}_{\ell})\subset Z(\Delta)$| that we will consider in the remaining purely group-theoretical lifting step). Actually, we are going to consider only characteristic subgroups $$T^{n}:=\lbrace t^{n}\; |\; t\in T\rbrace\subset T, \; n>0.$$ Lemma 2.7. Let |$G\subset{\rm GL}_{r}(\mathbb{F}_{\ell})$| be a subgroup and |$\Delta,T\subset G$| be two subgroups such that |$\Delta=\Delta^{+}$| is normal in |$G$| and (1) |$T$| is abelian (resp. abelian of prime-to-|$\ell$| order); (2) |$\Delta T $| is normal in |$G$|⁠. Then, for any integer |$n<\ell$| the subgroup |$T^{n}\subset T$| still satisfies properties (1), (2), and \[ [G:\Delta T^{n}]\leq [G:\Delta T]n^{r}. \] □ Proof. First, let us observe that for |$\ell>n$| one has |$[T:T^{n}]\leq n^{r}$|⁠. Indeed, write |$T =T^{(\ell)}\times T^{(\ell')}$| as the direct product of its |$\ell$|-part and prime-to-|$\ell$| part. As |$T^{(\ell')}$| acts semisimply, it is contained in a GL|$_{r}(\bar{\mathbb{F}}_{\ell})$|-conjugate of the torus |$\mathbb{G}_{m/\mathbb{F}_{\ell}}^{r}$|⁠. For |$\ell>n$|⁠, |$T/T^{n}$| has prime-to-|$\ell$| order hence |$T^{(\ell')}/(T^{(\ell')})^{n}\tilde{\rightarrow }T/T^{n}$| is a quotient of exponent |$\leq n$| of |$T^{(\ell')}$|⁠, which implies |$[T:T^{n}]\leq n^{r}$|⁠. Then, it remains to show that |$\Delta T^{n}$| is normal in |$G$|⁠. As |$\Delta T$| is normal in |$G$|⁠, it is actually enough to show that |$\Delta T^{n}$| is characteristic in |$\Delta T$|⁠. But this follows from the fact that for |$\ell> n$| one has $$\Delta T^{n}=\ker(\Delta T\twoheadrightarrow (\Delta T)^{ab}/n)$$ (use that |$\Delta=\Delta^{+}$|⁠). ■ 2.2.2.1 From |$c$| to |$c_{1,3}$| From |$c$| to |$c_{1,1}$|⁠. Set |$N :=\ker(\pi\circ c)\vartriangleleft G$|⁠. From Theorem 2.5, one has |$[G :N ]\leq 6^{r-1}$| hence a fortiori|$[T :T \cap N]\leq 6^{r-1}$|⁠. Thus, up to replacing |$T $| with |$T^{6^{r-1}!}\subset N\cap T \subset T $| and the constant |$d$| with |$(6^{r-1}!)^r d$|⁠, one may assume that |$c:T\rightarrow {\rm Aut}(\mathcal{D} )(\mathbb{F}_{\ell})$| factors through $$c_{1,1}:T\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell}).$$ From |$c_{1,1}$| to |$c_{1,2}$|⁠. We are to bound from above the cokernel of |$a$| by a constant depending only on |$r$|⁠. From the short exact sequence of flat sheaves $$1\rightarrow Z(\mathcal{D})\rightarrow \mathcal{D} \rightarrow \mathcal{D}/Z(\mathcal{D})\rightarrow 1$$ one obtains the exact sequence $$1\rightarrow Z(\mathcal{D})(\mathbb{F}_{\ell})\rightarrow \mathcal{D}(\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell}).$$ Claim:The image of |$\mathcal{D} (\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})$| has index |$\leq 2^{r-1}$|⁠. Proof of the claim. As |$Z(\mathcal{D} )$| is a finite commutative group scheme over |$\mathbb{F}_{\ell}$| it decomposes as the direct product $$ Z(\mathcal{D} )\simeq \mathcal{Z}^{\circ}\times \mathcal{Z}^{et} $$ of its connected and étale parts. Then, on the one hand, one has a short exact sequence of flat sheaves $$0\rightarrow \mathcal{Z}^{et}\rightarrow \mathcal{D}/\mathcal{Z}^{\circ}\rightarrow \mathcal{D}/Z(\mathcal{D})\rightarrow 0$$ hence an exact sequence $$0\rightarrow\mathcal{Z}^{et}(\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/\mathcal{Z}^{\circ})(\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})\rightarrow {\rm H}_{fl}^{1}(\mathbb{F}_{\ell},\mathcal{Z}^{et}).$$ Since |$\mathcal{Z}^{et}$| is étale over |$\mathbb{F}_{\ell}$|⁠, the right-hand term of this exact sequence can be computed in terms of Galois cohomology: $${\rm H}_{fl}^{1}(\mathbb{F}_{\ell},\mathcal{Z}^{et})={\rm H}^{1}(\mathbb{F}_{\ell},\mathcal{Z}^{et}(\overline{\mathbb{F}}_{\ell}))=\mathcal{Z}^{et}(\overline{\mathbb{F}}_{\ell})/(F-Id),$$ where the second equality follows from [33, Chapter XIII, Section 1, Proposition 1 and Remark] (here |$F$| denotes the Frobenius of the absolute Galois group of |$\mathbb{F}_{\ell}$|⁠). This shows that the order of |${\rm H}_{fl}^{1}(\mathbb{F}_{\ell},\mathcal{Z}^{et})$| divides |$|\mathcal{Z}^{et}(\overline{\mathbb{F}}_{\ell})|$|⁠. So, as the center of a rank |$\rho$| semisimple algebraic group has order at most |$2^{\rho}$| (see e.g., [32, p. 270]), the image of |$(\mathcal{D} /\mathcal{Z}^{\circ})(\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})$| has index |$\leq 2^{r-1}$|⁠. On the other hand, the morphism |$\mathcal{D} \rightarrow\mathcal{D} /\mathcal{Z}^{\circ}$| is radicial hence induces an isomorphism $$\mathcal{D}(\mathbb{F}_{\ell})\tilde{\rightarrow}(\mathcal{D}/\mathcal{Z}^{\circ})(\mathbb{F}_{\ell}).$$ This shows that |$\mathcal{D} (\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})$| and |$\mathcal{D} /\mathcal{Z}^{\circ}(\mathbb{F}_{\ell})\rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})$| have the same image. ■ So, up to replacing again |$T $| with |$T^{2^{r-1}!}\subset T $| and the constant |$(6^{r-1}!)^rd$| with |$(2^{r-1}!)^{r}(6^{r-1}!)^r d$|⁠, one may assume that |$c_{1,1}:T \rightarrow (\mathcal{D}/Z(\mathcal{D}))(\mathbb{F}_{\ell})$| factors through $$c_{1,2}:T\rightarrow \mathcal{D}(\mathbb{F}_{\ell})/Z(\mathcal{D})(\mathbb{F}_{\ell}).$$ From |$c_{1,2}$| to |$c_{1,3}$|⁠. We are to bound from above the cokernel of |$b$| by a constant depending only on |$r$|⁠. But this follows from Theorem 1.2 since for |$\ell\gg 0$| (depending on |$r$|⁠) one has |$\Delta =\mathcal{D}(\mathbb{F}_{\ell})^{+}$| hence $$[\mathcal{D}(\mathbb{F}_{\ell}):\Delta Z(\mathcal{D}(\mathbb{F}_{\ell}))]\leq [\mathcal{D}(\mathbb{F}_{\ell}):\mathcal{D}(\mathbb{F}_{\ell})^{+}]\leq 2^{r-1}.$$ So, up to replacing again |$T $| with |$T^{2^{r-1}!}\subset T $| and the constant |$(2^{r-1}!)^{r}(6^{r-1}!)^r d$| with |$(2^{r-1}!)^{2r}(6^{r-1}!)^r d$|⁠, one may assume that |$ c_{1,2}:T\rightarrow \mathcal{D}(\mathbb{F}_{\ell})/Z(\mathcal{D})(\mathbb{F}_{\ell})$| factors through $$c_{1,3}:T\rightarrow \Delta/\Delta\cap Z(\mathcal{D}(\mathbb{F}_{\ell})).$$ Note that |$Z:=Z(\mathcal{D})(\mathbb{F}_{\ell})\cap \Delta \subset Z(\mathcal{D})(\mathbb{F}_{\ell})$| has order |$\leq 2^{r-1}$|⁠. 2.2.2.2 End of the proof The end of the proof relies on the following two elementary group-theoretical lemmas. Lemma 2.8. Let |$G$| be a finite group, let |$Z\subset Z(G)$| be a central subgroup and let |$T$| be an abelian group. Consider an embedding problem of the form Then the induced embedding problem has a solution. □ Proof. Fix a set-theoretic lift |$\tilde{\psi}:T\rightarrow G$| of |$\overline{\psi}:T\rightarrow G/Z$| and write \begin{align} \gamma:\quad &T\times T &\rightarrow &\quad Z\\ &(t,t') &\rightarrow &\quad\tilde{\psi}(tt')\tilde{\psi}(t')^{-1}\tilde{\psi}(t)^{-1} \end{align} for the corresponding |$2$|-cocycle. Then, by definition one has \begin{align} \tilde{\psi}(tt')&=\gamma(t,t')\tilde{\psi}(t)\tilde{\psi}(t')\\ &=\tilde{\psi}(t't)=\gamma(t',t)\tilde{\psi}(t')\tilde{\psi}(t) \end{align} hence, with |$\omega(t,t'):=\gamma(t,t')^{-1}\gamma(t',t)$| one has $$\tilde{\psi}(t)\tilde{\psi}(t')=\omega(t,t')\tilde{\psi}(t')\tilde{\psi}(t).$$ With these notation, one obtains by a straightforward induction that $$\tilde{\psi}(tt')^{n}=\gamma(t,t')^{n}\omega(t',t)^{\frac{n(n-1)}{2}}\tilde{\psi}(t)^{n}\tilde{\psi}(t')^{n}.$$ Hence, setting |$N:=|Z|$|⁠, one obtains that |$\psi:=(-)^{2N}\circ \tilde{\psi}:T\rightarrow G$| is a group homomorphism lifting |$(-)^{2N}\circ\overline{\psi}=\overline{\psi}\circ (-)^{2N}:T\rightarrow G/Z$|⁠. ■ Lemma 2.9. Let |$G$| be a finite group, |$\Delta\subset G$| a subgroup and |$T$| an abelian group together with a group homomorphism |$\phi:T\rightarrow G$|⁠. Assume that we have a group homomorphism |$\psi:T\rightarrow \Delta$| such that |$\phi(t)\psi(t)^{-1}\in {\rm Cen}_{G}(\Delta)$| for all |$t\in T$|⁠. Then the set $$\Theta:=\lbrace \phi(t)\psi(t)^{-1}\; |\; t\in T\rbrace\subset G$$ is an abelian subgroup. □ Proof. Trivial. ■ It follows from Lemma 2.8 applied to the embedding problem that |$c_{1,3}\circ (-)^{2(2^{r-1}!)}:T\rightarrow \Delta/Z$| lifts to a morphism |$\psi :T \rightarrow \Delta $|⁠. By definition of |$c$|⁠, for every |$t\in T $| the element |$t^{2(2^{r-1}!)}\psi (t)^{-1}$| lies in Cen|$_{G}(\Delta)$| hence one can apply Lemma 2.9 to produce an abelian subgroup $$\Theta:=\lbrace t^{2(2^{r-1}!)}\psi (t)^{-1}\; |\; t\in T \rbrace\subset G,$$ which, by construction, lies in Cen|$_{G }(\Delta )$|⁠. Also, as |$\Delta \Theta =\Delta T^{2(2^{r-1}!)}$| one has $$[G :\Delta \Theta ]= [G :\Delta T^{2(2^{r-1}!)} ]\leq\delta(d,r):= 2^r(2^{r-1}!)^{3r}(6^{r-1}!)^rd$$ and |$\Delta \Theta $| is normal in |$G $| by Lemma 2.7 (for |$\ell>2(2^{r-1}!)^36^{r-1}!$|⁠). Remark 2.10. (Comparison with [31]) In order to remain in the rather elementary setting of [32], we deduce Corollary 2.3 from Theorems 2.2 and 2.1. Alternatively, we could also have deduced it from [31], which uses much more of the sophisticated machinery of the theory of algebraic groups. More precisely, [31, Theorem 02 (and its proof)]{LP} implies that there exists an integer |$\delta(r)\geq 1$| such that for every prime |$\ell$| and every finite subgroup |$G\subset {\rm GL}_{r}(\overline{\mathbb{F}}_{\ell})$|⁠, there exists an algebraic subgroup |$\mathcal{G}\hookrightarrow {\rm GL}_{r/\overline{\mathbb{F}}_{\ell}}$| such that |$[\mathcal{G}:\mathcal{G}^{\circ}]\leq \delta(r)$| and |$G\subset \mathcal{G}(\overline{\mathbb{F}}_{\ell})$|⁠, and there exist normal subgroups |$G_{3}\subset G_{2}\subset G_{1}\subset G$| with the following properties - |$G_{1}=G\cap \mathcal{G}^{\circ}(\overline{\mathbb{F}}_{\ell})$| (in particular |$[G:G_{1}]\leq \delta(r)$|⁠); - |$G_{1}/G_{2}$| is a direct product of simple groups of the form |$D(\mathcal{S}^{F})$|⁠, where |$\mathcal{G}^{\circ}\twoheadrightarrow \mathcal{S}$| is a simple quotient and |$F:\mathcal{S}\rightarrow\mathcal{S}$| a Frobenius map so that the derived subgroup |$D(\mathcal{S}^{F})$| be simple; - |$G_{2}/G_{3}$| embeds into the center of |$(\mathcal{G}^{\circ}/R_{u}(\mathcal{G}^{\circ}))(\overline{\mathbb{F}}_{\ell})$|⁠; - |$G_{3}=G\cap R_{u}(\mathcal{G}^{\circ})(\overline{\mathbb{F}}_{\ell})$|⁠. Let |$G^{+}\subset G$| denote the (normal) subgroup generated by the |$\ell$|-Sylow subgroups. One has |$G_{3}\subset G^{+}$| and, for |$\ell>\delta(r)$|⁠, one also has |$G^{+}\subset G_{1}$|⁠. As |$D(\mathcal{S}^{F})$| is simple of order divisible by |$\ell$|⁠, one obtains that |$G^{+}$| surjects on to |$G_{1}/G_{2}$|⁠. Assume that |$G^{+}$| acts semisimply on |$\overline{\mathbb{F}}_{\ell}^{\oplus r}$|⁠. Then as |$G_{3}$| is normal in |$G^{+}$|⁠, it also acts semisimply on |$\overline{\mathbb{F}}_{\ell}^{\oplus r}$| hence is trivial. Thus we may assume that |$\mathcal{G}^{\circ}$| is a connected reductive group and, then, the prime-to-|$\ell$| part |$T(\supset G_{2})$| of |$Z(G_{1})$| satisfies the conclusion of Corollary 2.3. Reference [31] actually implies Corollary 2.3 for finite subgroups of |$ \textrm{GL}_{r}(\overline{\mathbb{F}}_{\ell})$| (not only for subgroups of |$ {\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠). On the contrary, it is not clear whether the machinery of [31] can be exploited to show that Theorem 2.2 holds (when |$\ell$| divides |$|T|$|⁠) with |$H_{\ell}$| replaced by |$H_{\ell}\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}}_{\ell}$|⁠: the point is that the action of |$T$| by conjugation on |$\Delta$| will not extend, a priori, to an action on |$\mathcal{G}^{\circ}$|⁠. □ 3 Application to Representations of the étale Fundamental Group In this section, we carry out the proof of Theorem 1.1. So, let |$k$| be an algebraically closed field of characteristic |$p\geq 0$| and let |$X$| be a connected scheme, smooth and separated over |$k$|⁠. Let |$L$| be an infinite set of primes and let $$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L$$ be a family of |$\mathbb{F}_{\ell}$|-linear representations with |$r_{\ell}\leq r$| bounded as |$\ell$| varies. 3.1 Condition (F) Consider the following condition. (F) There exist a finitely generated field |$K$|⁠, an algebraically closed field |$\Omega$| containing |$K$|⁠, a curve |$C$| geometrically connected, smooth and separated over |$K$| and a morphism of profinite groups |$f:\pi_{1}(C_{\Omega})\rightarrow \pi_{1}(X)$| with open image such that for any |$\ell\in L$|⁠, |$\rho_{\ell}\circ f$| factors through the tame fundamental group |$\pi_{1}(C_{\Omega})\rightarrow \pi_{1}(C)\twoheadrightarrow \pi_{1}^{t}(C)$|⁠, that is, there exists a representation |$\pi_{1}^{t}(C)\rightarrow{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}) $| (again denoted by |$\rho_{\ell}$|⁠) such that the following diagram commutes The morphism |$\pi_{1}(C_{\Omega})\rightarrow \pi_{1}^{t}(C)$| can also be decomposed as |$\pi_{1}(C_{\Omega})\twoheadrightarrow\pi_{1}(C_{\overline{K}})\twoheadrightarrow\pi_{1}^{t}(C_{\overline{K}})\hookrightarrow \pi_{1}^{t}(C)$|⁠. Condition (F) may look technical but it is rather natural. It is there to ensure, on the one hand, that the |$\rho_{\ell}$|⁠, |$\ell\in L$| all factor through the same topologically finitely generated quotient (more precisely, setting |$N:=\cap\ker(\rho_{\ell})\vartriangleleft\pi_{1}(X)$|⁠, then |$\pi_{1}(C_{\Omega})\rightarrow\pi_{1}(X)/N$| has open image and factors through |$\pi_{1}(C_{\Omega})\twoheadrightarrow\pi_{1}^{t}(C_{\Omega})$|⁠, which is a topologically finitely generated profinite group) and, on the other hand, to reduce to the case where the base scheme is a curve (a situation where the étale fundamental group is better understood) over a finitely generated field (which will enable us to use arguments of arithmetic nature). Example 3.1. Condition (F) is always satisfied by families of representations arising from étale cohomology. More precisely, we have the following. Let |$k$| be a field, |$X$| a smooth and geometrically connected scheme over |$k$| and |$Y\stackrel{f}{\rightarrow}X$| a smooth, projective morphism. (In fact, |$f$| may be more general. For this generalization, see the proof of Corollary 4.6.) By the proper-smooth base change theorem, for any point |$x\in X$| one gets a family of |$\mathbb{F}_{\ell}$|-linear representations $$ \pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}(H^{{i}}(Y_{\overline{x}},\mathbb{F}_{\ell})),\; \ell\gg 0, $$ with |$r_{\ell}={\rm dim}_{\mathbb{F}_{\ell}}(H^{i}(Y_{\overline{x}},\mathbb{F}_{\ell}))$| independent of |$\ell$| for |$\ell\gg 0$|⁠. This follows from the fact that - |$H^{i}(Y_{\overline{x}},\mathbb{Z}_{\ell})$| is finitely generated over |$\mathbb{Z}_{\ell}$|⁠, and torsion free for |$\ell\gg 0$| [20], which, in turn, implies that |$H^{i}(Y_{\overline{x}},\mathbb{F}_{\ell})=H^{i}(Y_{\overline{x}},\mathbb{Z}_{\ell})/\ell$| for |$\ell\gg 0$| and; - the |$\mathbb{Q}_{\ell}$|-dimension of |$H^{i}(Y_{\overline{x}},\mathbb{Q}_{\ell})$| is independent of |$\ell$|⁠. (See [15, Section 2.4.1] for more details). Then, □ Proposition 3.2. The family |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}(H^{i}(Y_{\overline{x}},\mathbb{F}_{\ell}))$|⁠, |$\ell\gg 0$| satisfies condition (F). □ Proof. As all the objects are of finite type over |$k$|⁠, we may assume that |$f:Y\rightarrow X$| admits a model |$f_{0}:Y_{0}\rightarrow X_{0}$| over a finitely generated field |$k_{0}$|⁠. Up to replacing |$X_{0}$| by a nonempty open subscheme, one may assume that |$X_{0}$| is affine, hence quasi-projective. Fix an embedding |$X_{0}\hookrightarrow \mathbb{P}^{n}_{k_{0}}$| and let |$d$| denote the dimension of |$X$|⁠. Write |$Gr(d-1,n)$| for the Grassmanian of codimension |$d-1$| linear subspaces of |$\mathbb{P}^{n}_{k_{0}}$| and |$\xi$| for its generic point. Let |$\Omega$| be any algebraically closed field containing the residue field |$k(\xi_k)$| of the generic point |$\xi_{k}$| of |$Gr(d-1,n)_{k}$|⁠. Let $$Z:=\lbrace (x,V)\in X_{0}\times_{k_{0}}Gr(d-1,n)\; |\; x\in V\rbrace\hookrightarrow X_{0}\times_{k_{0}}Gr(d-1,n)$$ denote the incidence variety and |$p:Z\rightarrow Gr(d-1,n)$| the canonical projection. Note that |$Z_{\xi}\hookrightarrow X_{0,k_0(\xi)}$| is, by construction and Bertini’s theorem [26, Theorem 6.10, (2), (3)], a smooth, separated, geometrically connected curve over the residue field |$k_0(\xi)$| of |$\xi$|⁠, which is a finitely generated field. Claim: The morphism of étale fundamental groups $$ \pi_{1}(Z_{\xi,\Omega}){\,\rightarrow\,} \pi_{1}(X) $$ induced by |$Z_{\xi,\Omega}\stackrel{p_{\Omega}}{\rightarrow} X_{\Omega}\rightarrow X$| is an epimorphism. Proof of the claim. Indeed, for every connected étale cover |$\pi:X'\rightarrow X$| let again $$Z':=\lbrace (x',V)\in X'\times_{k}Gr(d-1,n)_{k}\; |\; \pi(x')\in V\rbrace=Z_{k}\times_{X}X'\hookrightarrow X'\times_{k}Gr(d-1,n)_{k}$$ denote the corresponding incidence variety. Then, from Bertini’s theorem, the generic fiber of the canonical projection |$p :Z'\rightarrow Gr(d-1,n)_k$| is smooth and geometrically connected. In particular, $$Z'\times_{Gr(d-1,n)_{k}}\Omega\rightarrow Z_k\times_{Gr(d-1,n)_{k}}\Omega=Z_{\xi,\Omega}$$ is a connected étale cover. This implies that $$\pi_{1}(Z_{\xi,\Omega})\rightarrow \pi_{1}(X)$$ is an epimorphism [22, Exp. V, Proposition 6.9]. Now, write |$K:=k_0(\xi)$| and |$C:=Z_{\xi}$|⁠. Fix |$c\in C_{\Omega}$|⁠, and let |$x$| be the image of |$c$| in |$X$| and |$\overline{x}$| a geometric point above |$c$|⁠. The family of |$\mathbb{F}_{\ell}$|-linear representations $$ \pi_{1}(C_{\Omega} )\rightarrow \pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}(H^{i}(Y_{\overline{x}},\mathbb{F}_{\ell})),\; \ell\gg 0, $$ then corresponds to the locally constant constructible sheaves |$R^{i}f_{C_{\Omega}*}\mathbb{F}_{\ell}$|⁠, where |$Y_{C_{\Omega}}\stackrel{f_{C_{\Omega}}}{\rightarrow}C_{\Omega}$| denote the pull-back of |$Y\stackrel{f}{\rightarrow}X$|via|$C_{\Omega} \rightarrow X$|⁠. But |$Y_{C_{\Omega}}\stackrel{f_{C_{\Omega}}}{\rightarrow}C_{\Omega}$| is also the pullback of |$Y_{0,K}\stackrel{f_{0,K}}{\rightarrow}X_{0,K}$|via|$C_{\Omega} \rightarrow C\rightarrow X_{0,K}$|⁠, which shows that the |$\rho_{\ell}$|⁠, |$\ell\gg 0$| factor through the arithmetic fundamental group |$\pi_{1}(C_{\Omega})\rightarrow\pi_{1}(C)$|⁠. The fact that up to replacing |$C$| with a connected étale cover the |$\rho_{\ell}$|⁠, |$\ell\gg 0$| factor through the tame fundamental group follows from de Jong’s alteration Theorem [1, Proposition 6.3.2] together with the facts that |$H^{i}(Y_{\overline{x}},\mathbb{Z}_{\ell})$| is torsion free and that |$H^{i}(Y_{\overline{x}},\mathbb{F}_{\ell})=H^{i}(Y_{\overline{x}},\mathbb{Z}_{\ell})/\ell$| for |$\ell\gg 0$|⁠. ■ 3.2 Proof of Theorem 1.1 Observe first that the second assertion of Theorem 1.1 follows from the fact that for a subgroup |$G\subset {\rm GL}_r(\mathbb{F}_{\ell})$|⁠, every proper subgroup |$H\subset G^{+}$| has index |$\geq \ell$|⁠. For the first assertion, let |$\pi_{1}(X)\stackrel{\rho_{\ell}^{ss}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$| denote the semi-simplification of |$\rho_{\ell}$| and set |$G_{\ell}:=$|im|$(\rho_{\ell})\subset{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$G_{\ell}^{ss}:=$|im|$(\rho_{\ell}^{ss})\subset{\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠. As the kernel of |$G_{\ell}\twoheadrightarrow G_{\ell}^{ss}$| is an |$\ell$|-group, it is enough to show that the conclusion of Theorem 1.1 holds for the $$\pi_{1}(X)\stackrel{\rho_{\ell}^{ss}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}),\; \ell\in L.$$ Furthermore, it follows from |$\ker(\rho_{\ell})\subset\ker(\rho_{\ell}^{ss})$| that condition (F) is still satisfied by the |$\pi_{1}(X)\stackrel{\rho_{\ell}^{ss}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$ \ell\in L$|⁠. So, without loss of generality, we may assume that |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$| is semisimple for every |$ \ell\in L$|⁠. By (and with the notation of) Condition (F), without loss of generality one may assume that |$k=\overline{K}$|⁠, that |$X=C_{\overline{K}}$| is a curve and that the |$\rho_\ell:\pi_1(C_{\overline{K}})\rightarrow {\rm GL}_{r_\ell}(\mathbb{F}_\ell)$|⁠, |$\ell\in L$| all factor through |$\pi_1(C_{\overline{K}})\twoheadrightarrow \pi_1^t(C_{\overline{K}})$|⁠. We proceed in two steps. Claim 1:There exists an open subgroup |$\Pi\subset \pi_{1}(C_{\overline{K}})$| such that for every open subgroup|$\Pi'\subset \Pi$| $$\rho_{\ell}(\Pi')=\rho_{\ell}(\Pi')^{+}Z(\rho_{\ell}(\Pi'))$$ for |$\ell\gg 0$| (depending on |$\Pi'$|⁠). For every |$\ell\in L$|⁠, let |$T_{\ell}\subset G_{\ell}$| be an abelian subgroup of prime-to-|$\ell$| order as in Corollary 2.3 that is, |$T_{\ell}$| centralizes |$G_{\ell}^{+}$|⁠, |$G_{\ell}^{+}T_{\ell}$| is normal in |$G_{\ell}$| and |$[G_{\ell}:G_{\ell}^{+}T_{\ell}]\leq \delta(r)$|⁠. (one has |$G_{\ell}\subset {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell}) \hookrightarrow {\rm GL}_{r}(\mathbb{F}_{\ell})$|⁠. As a topologically finitely generated profinite group has only finitely many open subgroups of bounded index, the set of the inverse images of the |$G_{\ell}^{+}T_{\ell}$|⁠, |$\ell\in L$| in |$\pi_{1}^t(C_{\overline{K}})$| is finite hence their intersection |$\Pi^t$| is again an open subgroup of |$\pi_1^t(C_{\overline{K}})$|⁠. Then, the inverse image |$\Pi$| of |$\Pi^t$| in |$\pi_{1}(C_{\overline{K}})$| is an open subgroup of |$\pi_{1}(C_{\overline{K}})$| and, by construction, it satisfies the conclusion of Claim 1. Claim 2:For every open subgroup |$\Pi\subset \pi_{1}(C_{\overline{K}})$| there exists an integer |$B_{\Pi}\geq 1$| such that for every prime |$\ell$| one has $$|Z(\rho_{\ell}(\Pi))| \leq B_{\Pi}.$$ The proof of Claim 2 can be reconstructed from the arguments involved in the proofs of [11, Proposition 3.1]. For the convenience of the reader, we recall the main steps. Without loss of generality, one may assume that |$\Pi=\pi_{1}(C_{\overline{K}})$|⁠. Write |$Z(G_{\ell})=:Z_{\ell}$|⁠. Then |$Z_{\ell}$| can be decomposed as the direct product of its |$\ell$| and prime-to-|$\ell$| part $$Z_{\ell}=Z_{\ell}^{(\ell)}\times Z_{\ell}^{(\ell')}.$$ As |$Z_{\ell}^{(\ell)}$| is characteristic in |$Z_{\ell}$| and |$Z_{\ell}$| is normal in |$G_{\ell}$|⁠, |$Z_{\ell}^{(\ell)}$| is normal in |$G_{\ell}$| and, in particular, it acts semisimply on |$H_{\ell}$|⁠. As it is an |$\ell$|-group, it also acts unipotently hence is necessarily trivial. So |$Z_{\ell}=Z_{\ell}^{(\ell')}$| is of prime-to-|$\ell$| order. Write |$F:=\mathbb{F}_{\ell}[Z_{\ell}]\subset {\rm End}(H_{\ell})$| for the (semisimple) |$\mathbb{F}_{\ell}$|-subalgebra generated by |$Z_{\ell}$| and set |$G_{\ell}^{ar}:=\rho_{\ell}(\pi_{1}(C))\vartriangleright G_{\ell}$|⁠. The action by conjugation of |$G_{\ell}^{ar}$| on |$G_{\ell}$| restricts to an action on the characteristic subgroup |$Z_{\ell}$|⁠, which in turn extends by |$\mathbb{F}_{\ell}$|-linearity to an action on |$F$|⁠. As |$G_{\ell}$| acts trivially on |$Z_{\ell}$|⁠, we thus get where |$F\otimes_{\mathbb{F}_{\ell}}\overline{\mathbb{F}}_{\ell}\simeq \overline{\mathbb{F}}_{\ell}^{ t}$| as an |$\overline{\mathbb{F}}_{\ell}$|-algebra (since |$\mathbb{F}_{\ell}$| is perfect). So, there exists a finite extension |$K_{\ell}$| of |$K$| with degree |$[K_{\ell}:K]\leq t!\leq r_{\ell}!\leq r!$| such that |$Z_{\ell}\subset Z(\rho_{\ell}(\pi_{1}(C_{K_{\ell}})))$|⁠. Write |$\overline{Z}_{\ell}$| for the image of |$Z_{\ell}$| in the abelianization |$G_{\ell}\twoheadrightarrow G_{\ell}^{ab}$|⁠. One has $$Z_{\ell}\twoheadrightarrow \overline{Z}_{\ell}\hookrightarrow G_{\ell}^{ab} \twoheadleftarrow \pi_{1}^{t}(C_{\overline{K}})^{ab} \twoheadleftarrow \pi_{1}(C_{\overline{K}})^{ab} $$ as |$\Gamma_{K}$|-modules. Note that |$Z_{\ell}$| and |$\overline{Z}_{\ell}$| are trivial as |$\Gamma_{K_{\ell}}$|-modules. From the arguments used in Claim 3.4 in the proof of [11, Proposition 3.1], it is enough to bound |$|\overline{Z}_{\ell}|$| independently of |$\ell$|⁠. For this, we reduce by specialization to the case where |$K$| is finite. Up to enlarging |$K$|⁠, we may assume that |$C$| admits a (unique) smooth compactification |$C\subset C^{cpt}$| with |$C^{cpt}\setminus C$| étale over |$K$| and that |$C$| has a |$K$|-rational point |$c$|⁠. Consider a model |$[{\rm spec}(R)\overset{c}\rightarrow\mathcal{C}\subset\mathcal{C}^{cpt} \rightarrow {\rm spec}(R)]$| of |$[{\rm spec}(K)\overset{c}\rightarrow C\subset C^{cpt}\rightarrow {\rm spec}(K)]$|⁠. More precisely, |$R$| is a finitely generated normal integral |$\mathbb{Z}$|-algebra with fraction field |$K$|⁠; |$\mathcal{C}^{cpt}\rightarrow {\rm spec}(R)$| is a proper, smooth, geometrically connected curve over |$R$| and |$\mathcal{C}^{cpt}\setminus \mathcal{C}$| is a relatively finite étale divisor, such that |$\mathcal{C}^{cpt}\times_{R}K $| and |$\mathcal{C}\times_{R}K$| are isomorphic to (and will be identified with) |$C^{cpt}$| and |$C$| respectively over |$K$|⁠; and |$c:{\rm spec}(R)\to\mathcal{C}$| is an (a unique) extension of |$c: {\rm spec}(K)\to C$| (under the identification |$\mathcal{C}\times_{R}K =C$|⁠). Fix any closed point |$v\in {\rm spec}(R)$| and let |$p>0$| denote its residue characteristic. Then one gets a specialization isomorphism for the prime-to-|$p$| part of the étale fundamental groups (see [22]) $$ \pi_{1}^{t}(C_{\overline{K}})^{(p')} = \pi_{1}(C_{\overline{K}})^{(p')} \tilde{\rightarrow} \pi_{1}(\mathcal{C}_{\overline{v}})^{(p')} = \pi_{1}^{t}(\mathcal{C}_{\overline{v}})^{(p')} , $$ which induces an isomorphism on the abelianization $$ \pi_{1}^{t}(C_{\overline{K}})^{ab,(p')} = \pi_{1}(C_{\overline{K}})^{ab,(p')} \tilde{\rightarrow} \pi_{1}(\mathcal{C}_{\overline{v}})^{ab,(p')} = \pi_{1}^{t}(\mathcal{C}_{\overline{v}})^{ab,(p')}. $$ This isomorphism is compatible with the actions of $$\Gamma_{K}\supset D_{v}\twoheadrightarrow \Gamma_{\kappa(v)},$$ where |$D_{v}$| stands for the decomposition group at |$v$| and |$\kappa(v)$| for the residue field at |$v$|⁠. Further, let |$R_{\ell}$| be the integral closure of |$R$| in |$K_{\ell}$| and let |$v_{\ell}$| be the closed point of |${\rm spec}(R_{\ell})$| above |$v$| such that |$D_{v_{\ell}}\subset D_v$|⁠. Now, one gets homomorphisms $$\overline{Z}_{\ell}^{(p')}\hookrightarrow G^{ab,(p')}_{\ell}\twoheadleftarrow \pi_{1}^{t}(C_{\overline{K}})^{ab,(p')} \tilde{\rightarrow} \pi_{1}^{t}(\mathcal{C}_{\overline{v}})^{ab,(p')}, $$ which are compatible with the actions of |$\Gamma_{K_{\ell}}\supset D_{v_{\ell}}\twoheadrightarrow \Gamma_{\kappa(v_{\ell})}$|⁠. In particular, the action of |$D_{v_{\ell}}$| on |$G^{ab,(p')}_{\ell}$| factors through |$\Gamma_{\kappa(v_{\ell})}$|⁠, as |$G_{\ell}^{ab,(p')}$| is a quotient of the |$\Gamma_{\kappa(v_{\ell})}$|-module |$\pi_{1}^{t}(\mathcal{C}_{\overline{v}})^{ab,(p')}.$| Note that $$[\Gamma_{\kappa(v)}:\Gamma_{\kappa(v_{\ell})}]\leq [D_{v}:D_{v_{\ell}}]\leq [\Gamma_{K}:\Gamma_{K_{\ell}}]\leq r!. $$ Since |$\Gamma_{\kappa(v)}\simeq\hat{\mathbb{Z}}$| is a finitely generated profinite group, the intersection |$\Gamma$| of all open subgroups |$\Gamma'\subset \Gamma_{\kappa(v)}$| with |$[\Gamma_{\kappa(v)}:\Gamma']\leq r!$| is again an open subgroup. (The index |$[\Gamma_{\kappa(v)}: \Gamma]$| is equal to the least common multiple of |$1,\dots, r!$|⁠, which is independent of |$\ell$|⁠.) Write |$\kappa$| for the finite extension of |$\kappa(v)$| corresponding to |$\Gamma\subset\Gamma_{\kappa(v)}$|⁠, and let |$\phi$| denote the |$|\kappa|$|-th power Frobenius element, which is a generator of |$\Gamma=\Gamma_{\kappa}$|⁠. By construction, |$\phi$| acts trivially on |$\overline{Z}_{\ell}^{(p')}$|⁠. This implies that |$|\overline{Z}_{\ell}^{(p')} |$| is bounded from above by |$|P_{\phi}(1)|$|⁠, where |$P_{\phi}(T) \in\mathbb{Z}[T](\subset \prod_{a\neq p }\mathbb{Z}_{a}[T]) $| is the characteristic polynomial of |$\phi$| acting on |$\pi_{1}^{t}(\mathcal{C}_{\overline{v}})^{ab,(p')}$| by conjugation. (The proof of this fact can be seen in the last part of [11, Proposition 3.1].) Note that |$P_{\phi}(1)$| is a nonzero integer which is independent of |$\ell$|⁠. To treat the |$p$|-part |$\overline{Z}_{\ell}^{(p)}$|⁠, we proceed as follows. If |$K$| has characteristic |$0$|⁠, consider the nonempty open subscheme |$V:={\rm spec}(R[1/p])\subset{\rm spec}(R)$|⁠. If |$K$| has characteristic |$>0$|⁠, then the characteristic of |$K$| must coincide with |$p$|⁠. In this case, consider the |$p$|-rank of (jacobian varieties of) curves obtained as fibers of the family |$\mathcal{C}^{cpt}\rightarrow {\rm spec}(R)$|⁠, and let |$V\subset {\rm spec}(R)$| denote the nonempty open subscheme on which the fiber has maximal |$p$|-rank. Now, fix any closed point |$w\in V\subset {\rm spec}(R)$| and let |$q>0$| denote its residue characteristic. (Thus, if the characteristic of |$K$| is |$0$| (resp. |$>0$|⁠), one has |$q\neq p$| (resp. |$q=p$|⁠).) Then one gets a specialization isomorphism for the pro-|$p$| part of the étale fundamental groups $$ \pi_{1}^{t}(C_{\overline{K}})^{(p)} \,\tilde{\rightarrow}\, \pi_{1}^{t}(\mathcal{C}_{\overline{w}})^{(p)}. $$ (When the characteristic of |$K$| is |$0$|⁠, again this can be seen in [22]. When the characteristic of |$K$| is |$p>0$|⁠, see, e.g., [4]). Now, we can bound the |$p$|-part |$|\overline{Z}_{\ell}^{(p)}|$| just as in the case of the prime-to-|$p$| part |$|\overline{Z}_{\ell}^{(p')}|$|⁠. Due to the revision of [15] and an improvement of what was formally [15, Thm. 2.8], this reference is no longer accurate. To conclude the proof of (the first assertion of) Theorem 1.1, one just applies again the finiteness argument in the proof of Claim 1 with |$G_{\ell}^{+}$| instead of |$G_{\ell}^{+}T_{\ell}$|⁠. 3.3 The semisimple case We end this section with the following refinements of Theorem 1.1 when the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$\ell\in L$| are furthermore assumed to be semisimple. Corollary 3.3. Assume that Condition (F) is satisfied and that |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$| is semisimple for every |$ \ell\in L$|⁠. Then there exists an open subgroup |$\Pi\subset \pi_{1}(X)$| such that for every open subgroup |$\Pi'\subset \Pi$| $$\rho_{\ell}(\Pi')=\rho_{\ell}(\Pi')^{+}\;{\rm and}\;\rho_{\ell}(\Pi')^{ab}=0$$ for |$\ell\gg 0$| (depending on |$\Pi'$|⁠). □ Proof. This follows directly from Theorem 1.1 and Lemma 3.4 below. ■ Lemma 3.4. For |$\ell\gg 0$| (depending only on |$r$|⁠) and every subgroup |$G\subset {\rm GL}_{r}(\mathbb{F}_{\ell})$| such that |$G=G^{+}$| and |$G$| acts semisimply on |$H_{\ell}=\mathbb{F}_{\ell}^{\oplus r}$| one has \[ G^{ab}=0. \] □ Proof. As |$G=G^{+}$| and for |$\ell> r$|⁠, |$G$| contains no element of order |$\ell^{2}$|⁠, one can identify |$G^{ab}$| with the dual of Hom|$(G,\mathbb{F}_{\ell})={\rm H}^{1}(G,\mathbb{F}_{\ell})$| (where |$\mathbb{F}_{\ell}$| denotes the trivial |$G$|-module). From the split short exact sequence of |$G$|-modules $$0\rightarrow\mathbb{F}_{\ell}\rightarrow \mathbb{F}_{\ell}\oplus H_{\ell}\rightarrow H_{\ell}\rightarrow 0,$$ one obtains an embedding of abelian groups $$ {\rm H}^{1}(G,\mathbb{F}_{\ell})\hookrightarrow {\rm H}^{1}(G,\mathbb{F}_{\ell}\oplus H_{\ell}).$$ But as |$\mathbb{F}_{\ell}\oplus H_{\ell}$| is a faithful semisimple |$G$|-module, it follows from [32, Theorem E] that |$ {\rm H}^{1}(G,\mathbb{F}_{\ell}\oplus H_{\ell})=0$| for |$\ell\gg 0$| (depending only on |$r$|⁠). ■ Corollary 3.5. Assume that |$k=\overline{k}_{0}$| with |$k_{0}$| a finite field and that |$X=X_{0,k}$| with |$X_{0}$| a scheme geometrically connected, smooth and separated over |$k_{0}$|⁠. Also, assume that the |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$ \ell\in L$| are semisimple and satisfy condition (F). Then there exists an open subgroup |$\Pi\subset \pi_{1}(X_{0})$| such that for every open subgroup |$\Pi'\subset \Pi$| and prime |$\ell\gg 0$| (depending on |$\Pi'$|⁠), one has $$Z(\rho_{\ell}(\Pi'))\rho_{\ell}(\Pi'\cap\pi_{1}(X))=\rho_{\ell}(\Pi')$$ and |$\rho_{\ell}(\Pi'\cap\pi_{1}(X))=\rho_{\ell}(\Pi'\cap\pi_{1}(X))^{+}$|⁠.□ Proof. From Theorem 1.1, up to replacing |$X$| with a connected étale cover one may assume that |$\rho_{\ell}( \pi_{1}(X))=\rho_{\ell}( \pi_{1}(X))^{+}$| for |$\ell\gg 0$| and that for all open subgroup |$\Pi\subset \pi_{1}(X_{0})$| and all prime |$\ell\gg 0$| (depending on |$\Pi$|⁠), one has |$ \rho_{\ell}(\Pi\cap\pi_{1}(X))=\rho_{\ell}( \pi_{1}(X))$|⁠. Set |$G_{\ell}:=\rho_{\ell}( \pi_{1}(X_{0}))$|⁠, |$\Delta_{\ell}:=\rho_{\ell}( \pi_{1}(X))$| and |$C_{\ell}:=G_{\ell}/\Delta_{\ell}$|⁠. Fix |$\phi_{\ell}\in G_{\ell}$| lifting any generator of |$C_{\ell}$| and set |$T_{\ell}:=\langle\phi_{\ell}\rangle$|⁠. Then Theorem 2.2 applied to |$G=G_{\ell}$|⁠, |$\Delta=\Delta_{\ell}$| and |$T=T_{\ell}$| provides an element |$\varphi_{\ell}\in G_{\ell}$| commuting with |$\Delta_{\ell}$| and mapping to a generator of a subgroup of index |$\leq \delta(1,r)$| of the cyclic group |$C_{\ell}$|⁠. So, up to replacing |$k_{0}$| by its degree |$\delta(1,r)!$| extension, we are done. ■ Corollary 3.3 extends [19, Proposition 16] (which resorts to the delicate techniques developed by Serre to describe the algebraic envelope of the Galois image on |$\ell$|-torsion points of abelian varieties over number fields [34]—see [19, Theorems 17, 18, and App. B]) to arbitrary base fields |$k$| and arbitrary families of semisimple representations |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$|⁠, |$\ell\in L$|⁠. In particular, the results of [19] essentially extend as they are to families of representations arising from étale cohomology in characteristic |$0$|⁠. We refer to [14] (where Corollary 3.5 is used) for extensions of the results of [19] to families of representations arising from étale cohomology in arbitrary characteristic. Remark 3.6. (1) (About Claim 2 in the proof of Theorem 1.1) Though we will not need it in the remaining part of this article, let us mention that the arguments used in the proof of “Claim 3.2 implies Proposition 3.1” in the proof of [11, Proposition 3.1] yield the following seemingly stronger version of Claim 2.} Assume that condition (F) is satisfied and that |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow} {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell})$| is semisimple for every |$ \ell\in L$|⁠. Then, for every open subgroup |$\Pi\subset \pi_{1}(X)$| there exists an integer |$B_{\Pi}\geq 1$| such that for every prime |$\ell$| and abelian normal subgroup |$Z_{\ell}\vartriangleleft \rho_{\ell}(\Pi)$| one has $$|Z_{\ell}| \leq B_{\Pi}.$$ (2) From Remark 2.10 (and [24, Theorem C] instead of Lemma 3.4), Theorem 1.1 (resp. Corollary 3.3) remains true for families of continuous (resp. semisimple continuous) representations |$\pi_{1}(X)\stackrel{\rho_{\ell}}{\rightarrow}{\rm GL}_{r_{\ell}}(\overline{\mathbb{F}}_{\ell}),\; \ell\in L$| satisfying Condition (F). □ 4 Applications to Almost |$\boldsymbol{\ell}$|-Independence in the Sense of Serre 4.1 Almost |$\boldsymbol{\ell}$|-independence in the sense of Serre Let us first recall the definition of almost |$\ell$|-independence for a family of |$\ell$|-adic representations of a profinite group. Let |$\Gamma$| be a profinite group, |$L$| an infinite set of primes, |$\Gamma_{\ell}$|⁠, |$\ell\in L$| a family of |$\ell$|-adic Lie groups and $$\rho_{\ell}:\Gamma\rightarrow \Gamma_{\ell},\; \ell\in L$$ a family of continuous representations. (We will refer to such a family as a family of |$\ell$|-adic representations of |$\Gamma$| for short). One says that the |$\rho_{\ell}$|⁠, |$\ell\in L$| are (⁠|$\ell$|-)independent if the resulting product representation $$\rho:=(\rho_{\ell})_{\ell\in L}:\Gamma\rightarrow \prod_{\ell\in L}\Gamma_{\ell}$$ satisfies $$\rho(\Gamma)=\prod_{\ell\in L}\rho_{\ell}(\Gamma).$$ and that the |$\rho_{\ell}$|⁠, |$\ell\in L$| are almost (⁠|$\ell$|-)independent if there exists an open subgroup |$\Pi\subset\Gamma$| such that the |$\rho_{\ell}|_{\Pi}$|⁠, |$\ell\in L$| are independent. The notion of (almost) |$\ell$|-independence was introduced by Serre [36] (see also [35]); it corresponds to the case where the image of the product representation |$\rho:=(\rho_{\ell})_{\ell\in L}:\Gamma\rightarrow \prod_{\ell\in L}\Gamma_{\ell}$| is as large as one can (reasonably) expect. The main result of [36] is a criterion for almost |$\ell$|-independence when |$\Gamma$| is the absolute Galois group of a number field; this criterion applies in particular to show. Theorem 4.1. [36, Section 3.2] Let |$k$| be a number field and let |$Y$| be a scheme separated and of finite type over |$k$|⁠. Then the representations $$\rho_{\ell}:\Gamma_{k}\rightarrow {\rm GL}({\rm H}^{i}(Y_{\overline{k}},\mathbb{Q}_{\ell})),\; \ell:\;{\rm prime}$$ are almost-independent (here |${\rm H}^{i}(-,\mathbb{Q}_{\ell})$| may refer either to the usual |$\ell$|-adic cohomology or to the |$\ell$|-adic cohomology with compact support). □ The proof of Serre’s criterion is built on several intermediate technical results, some of which we will re-use below and recall now. Lemma 4.2. [36, Lemmas 1 and 3] (1) If for |$\ell\not=\ell'$| no simple quotient of |$\rho_{\ell}(\Gamma)$| is isomorphic to a simple quotient of |$\rho_{\ell'}(\Gamma)$| then the |$\rho_{\ell}$|⁠, |$\ell\in L$| are independent. (2) If there exists a finite subset |$F\subset L$| such that the |$\rho_{\ell}$|⁠, |$\ell\in L\setminus F$| are independent then the |$\rho_{\ell}$|⁠, |$\ell\in L$| are almost independent. □ For every prime |$\ell$|⁠, let |$\Sigma_{\ell}$| denote the set of all (isomorphism classes of) finite groups which are either a simple group of Lie type in characteristic |$\ell$| (see [36, Section 6.1]) or |$\mathbb{Z}/\ell$|⁠. Theorem 4.3. [36, Theorems 4 and 5] (1) Every finite simple subquotient of |${\rm GL}_{r}(\mathbb{Z}_{\ell})$| of order divisible by |$\ell$| is in |$\Sigma_{\ell}$| for |$\ell\gg 0$| (depending on |$r$|⁠). (2) For |$\ell,\ell'\geq 5$|⁠, |$\ell\not=\ell'$| one has |$\Sigma_{\ell}\cap \Sigma_{\ell'}=\emptyset$|⁠. □ 4.2 Notation Let |$k$| be a field of characteristic |$p\geq 0$| and let |$X$| be a scheme geometrically connected, smooth and separated over |$k$|⁠. Let |$L$| be an infinite set of primes and consider a family of |$\ell$|-adic representations together with their reduction modulo |$\ell$| with |$r_{\ell}\leq r$| bounded as |$\ell$| varies. Write \begin{align} G_{\ell^{\infty}}&:={\rm{im}}(\rho_{\ell^{\infty}});&G_{\ell}&:={\rm{im}}(\rho_{\ell});\\ G_{\ell^{\infty}}^{geo}&:=\rho_{\ell^{\infty}}(\pi_{1}(X_{\overline{k}}))\vartriangleleft G_{\ell^{\infty}};& G_{\ell}^{geo}&:=\rho_{\ell}(\pi_{1}(X_{\overline{k}}))\vartriangleleft G_{\ell}. \end{align} 4.3 Almost |$\ell$|-independence for families of |$\ell$|-adic representations of the étale fundamental group Corollary 4.4. Assume that |$k$| is algebraically closed and that the |$\rho_{\ell}$|⁠, |$\ell\in L$| satisfy condition (F). Then the |$\rho_{\ell^{\infty}}$|⁠, |$\ell\in L$| are almost independent. □ Proof. From Theorem 1.1, Theorem 4.3 (1) and Lemma 4.2 (2), up to replacing |$X$| by a connected étale cover and excluding finitely many |$\ell\in L$|⁠, one may assume that (1) |$2,3\notin L$|⁠; (2) for every |$\ell\in L$|⁠, every finite simple subquotient of |${\rm GL}_{r}(\mathbb{Z}_{\ell})$| of order divisible by |$\ell$| is in |$\Sigma_{\ell}$|⁠; (3) |$G_{\ell}=G_{\ell}^{+}$|⁠. In particular, from (3) every simple quotient of |$G_{\ell}$| has order divisible by |$\ell$|⁠. Since |$G_{\ell^{\infty}}$| is an extension of |$G_{\ell}$| by the kernel of reduction modulo |$\ell$|⁠, which is a pro-|$\ell$| group, it follows from (2) that every finite simple quotient of |$G_{\ell^{\infty}}$| lies in |$\Sigma_{\ell}$|⁠. The conclusion thus follows from Theorem 4.3 (2) and Lemma 4.2 (1). ■ Recall that every closed point |$x\in X$| viewed as a morphism |$x:{\rm spec}(k(x))\rightarrow X$| induces a quasi-section of the fundamental short exact sequence for |$\pi_{1}(X)$| Write \begin{align} \rho_{\ell^{\infty},x}&:= \rho_{\ell^{\infty}}\circ\sigma_{x}:\Gamma_{k(x)}\rightarrow {\rm GL}_{r_{\ell}}(\mathbb{Z}_{\ell});& \rho_{\ell,x}&:= \rho_{\ell}\circ\sigma_{x}:\Gamma_{k(x)}\rightarrow {\rm GL}_{r_{\ell}}(\mathbb{F}_{\ell});\\ G_{\ell^{\infty},x}&:={\rm{im}}(\rho_{\ell^{\infty},x});& G_{\ell,x}&:={\rm{im}}(\rho_{\ell,x}).\\ \end{align} The following extends [21, Theorem 3.4], which only works for |$k$| a number field, to an arbitrary base field |$k$|⁠. Corollary 4.5. Assume that the |$\rho_{\ell}|_{\pi_{1}(X_{\overline{k}})}$|⁠, |$\ell\in L$| satisfy condition (F) and that there exists a closed point |$x\in X$| such that the |$\rho_{\ell^{\infty},x}$|⁠, |$\ell\in L$| are almost independent then the |$\rho_{\ell^{\infty}}$|⁠, |$\ell\in L$| are almost independent. □ Proof. Up to replacing |$k$| by a finite extension, one may assume that |$x\in X(k)$|⁠. From Corollary 4.4, the |$\rho_{\ell^{\infty}}|_{\pi_{1}(X_{\overline{k}})}$|⁠, |$\ell\in L$| are almost independent. Hence, up to replacing |$X$| by a connected étale cover one may assume that both the |$\rho_{\ell^{\infty}}|_{\pi_{1}(X_{\overline{k}})}$|⁠, |$\ell\in L$| and the |$\rho_{\ell^{\infty},x}$|⁠, |$\ell\in L$| are independent. As |$\pi_{1}(X)=\pi_{1}(X_{\overline{k}})\rtimes_{\sigma_{x}}\Gamma_{k}$| it straightforwardly follows that the |$\rho_{\ell}$|⁠, |$\ell\in L$| are independent as well. ■ 4.4 Almost |$\ell$|-independence for motivic families of |$\ell$|-adic Galois representations Let |$K$| be a field of characteristic |$p\geq 0$| and |$Y$| a scheme separated and of finite type over |$K$|⁠. Consider the resulting family of |$\ell$|-adic Galois representations $$ \rho_{\ell^{\infty}}:\Gamma_{K}\rightarrow {\rm GL}({\rm H}^{i}(Y_{\overline{K}},\mathbb{Q}_{\ell})),\; \ell:\;{\rm prime}\not= p, $$ where |${\rm H}^{i}(-,\mathbb{Q}_{\ell})$| may refer either to the usual |$\ell$|-adic cohomology or to the |$\ell$|-adic cohomology with compact support. As in [21], one can apply the uniformity results of [24] (see also [27]) to reduce the case of |$\ell$|-adic Galois representations to the case of |$\ell$|-adic representations of the étale fundamental group. Corollary 4.6. Assume that |$K$| is finitely generated over |$k$| and that |$k$| is either |$\mathbb{Q}$| or an algebraically closed field. Then the |$\ell$|-adic Galois representations $$\rho_{\ell^{\infty}}:\Gamma_{K}\rightarrow {\rm GL}({\rm H}^{i}(Y_{\overline{K}},\mathbb{Q}_{\ell}))$$ are almost independent. □ Proof. Let |$X$| be a scheme smooth, separated and geometrically connected over |$k$| and whose generic point |$\eta$| has residue field |$k(\eta)=K$| and let |$\mathcal{Y}\stackrel{f}{\rightarrow} X$| be a morphism of |$k$|-schemes, separated and of finite type such that Let |$R^{i}f\mathbb{Z}_{\ell}$| denote either |$R^{i}f_{*}\mathbb{Z}_{\ell}$| or |$R^{i}f_{!}\mathbb{Z}_{\ell}$|⁠, corresponding to the choice that |${\rm H}^{i}(-,\mathbb{Q}_{\ell})$| is the usual |$\ell$|-adic cohomology or the |$\ell$|-adic cohomology with compact support. From [24, Corollary 2.6], there exists a dense open subscheme |$U\hookrightarrow X$| such that |$R^{i}f\mathbb{Z}_{\ell}|_{U}$| is lisse and of formation compatible with any base change. In particular |${\rm H}^{i}(Y_{\overline{K}},\mathbb{Q}_{\ell}) ={\rm H}^{i}(\mathcal{Y}_{\overline{\eta}},\mathbb{Q}_{\ell}) =(R^{i}f\mathbb{Q}_{\ell})_{\overline{\eta}} $| and |$\rho_{\ell^{\infty}}$| factors through |$\Gamma_{K}\twoheadrightarrow\pi_{1}(U)$|⁠. As in Example 3.1, resorting to [26, Theorem 6.10], one can construct a finitely generated field extension |$L$| of |$k$| and a curve |$C\subset U_{L} $| smooth, separated and geometrically connected over |$L$|⁠, such that the induced morphism |$\pi_{1}(C_{\overline{L}})\twoheadrightarrow\pi_{1}(U_{\overline{k}})$| is surjective and that $$\pi_{1}(C_{\overline{L}})\twoheadrightarrow\pi_{1}(U_{\overline{k}})\stackrel{\rho_{\ell^{\infty}}}{\rightarrow}{\rm GL}({\rm H}^{i}(\mathcal{Y}_{\overline{\eta}},\mathbb{Q}_{\ell}))$$ factors through |$\pi_{1}(C_{\overline{L}})\rightarrow \pi_{1}(C)$|⁠. From de Jong’s alteration theorem [1, Proposition 6.3.2], up to replacing |$C$| with a connected étale cover, we may assume that the resulting representation |$\pi_{1}(C)\rightarrow {\rm GL}({\rm H}^{i}(\mathcal{Y}_{\overline{\eta}},\mathbb{Q}_{\ell}))$| factors through the tame fundamental group |$\pi_{1}(C)\twoheadrightarrow\pi_{1}^{t}(C)$|⁠. Now, let |$\rho_{\ell}$| denote the modulo |$\ell$| representation obtained as the reduction of the |$\ell$|-adic representation |$\rho_{\ell^{\infty}}:\Gamma_{K}\rightarrow {\rm GL}(\overline{{\rm H}}^{i}(Y_{\overline{K}},\mathbb{Z}_{\ell}))\subset {\rm GL}({\rm H}^{i}(Y_{\overline{K}},\mathbb{Q}_{\ell}))$|⁠, where |$\overline{{\rm H}}^{i}(Y_{\overline{K}},\mathbb{Z}_{\ell}):= {\rm H}^{i}(Y_{\overline{K}},\mathbb{Z}_{\ell})/({\rm torsion})$|⁠. Again, as in Example 3.1, this implies that the |$\rho_{\ell}$| satisfy condition (F) for |$\ell\gg 0$|⁠. (Note that |$r_{\ell}:= {\rm dim}_{\mathbb{F}_{\ell}}\overline{{\rm H}}^{i}(Y_{\overline{K}},\mathbb{Z}_{\ell})/\ell ={\rm dim}_{\mathbb{Q}_{\ell}}{\rm H}^{i}(Y_{\overline{K}},\mathbb{Q}_{\ell})$| is bounded as |$\ell$| varies. See [24, Corollary 1.3].) The conclusion now follows from Corollary 4.5 and Theorem 4.1 (when |$k=\mathbb{Q}$|⁠) or Corollary 4.4 (when |$k$| is algebraically closed). ■ For a different approach (following more closely Serre’s original strategy in [36]) of Corollary 4.6 see [21] (when |$k=\mathbb{Q}$|⁠) and the preprint [2] (when |$k$| is algebraically closed). References 1 Berthelot P. “Altération des variétés algébriques (d’après A. de Jong).” In Sém. Bourbaki 1995/1996 , vol. 815 . Astérisque, 241 . Paris : S.M.F. , 1997 . 2 Böckle G. , Gajda W. and Petersen. S. “Independence of |$\ell$| and the geometric Galois action on étale cohomology.” J. Reine Angew. Math . ( forthcoming ). DOI: https://doi.org/10.1515/crelle-2015-0024. Borel A. Linear Algebraic Groups . New York-Amsterdam : WA Benjamin Advanced Book Program , 1969 . Bouw I. “The |$p$|-rank of Curves and Covers of Curves.” In Courbes semi-stables et groupe fondamental en gómétrie algébrique, (Luminy, 1998) . 1998 , 267 – 77 . 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Cornulier,, Yves

doi: 10.1093/imrn/rnx187pmid: N/A

Abstract Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate. After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent groups for which this is not equivalent to the volume growth. In particular, we provide an example for which the degree of growth is not an integer; it has dimension 7. Finally, we gather some open questions. 1 Introduction 1.1 Background Every locally compact group |$G$| has a left Haar measure |$\mu$|⁠, unique up to positive scalar multiplication. If in addition |$G$| is generated by a symmetric compact neighborhood |$S$| of 1, the function |$b(n)=\mu(S^n)$| is called the volume growth (or word growth) of |$G$|⁠; while its values depend on the choice of |$(S,\mu)$|⁠, its asymptotics (in the usual meaning, recalled in Section 3.1) does not. The volume growth is either exponential or subexponential. Those compactly generated locally compact group with polynomially bounded growth have been characterized by Guivarch and Jenkins [13, 14] in the case of connected Lie groups, Gromov in the case of discrete groups [11], and Losert [15] in general. All such groups are commable, and hence quasi-isometric, to simply connected nilpotent Lie groups, and thus, by work of Guivarch [13] have an integral degree of polynomial growth that is easily computable in terms of the Lie algebra structure (see Section 2.1). The object of study of the article is the following related notion of growth, introduced by Gromov in [12, p. 333]. Definition 1.1. Let |$H$| be a locally compact group and |$|\cdot|$| the word length relative to some choice of compact generating subset. If |$X\subset H$|⁠, define its systole as |${\mathrm{sys}}(X)=\inf\{|x|:x\in X\smallsetminus\{1\}\}\in\mathbf{R}_+\cup\{+\infty\}$|⁠. Endow |$H$| with some left-invariant Haar measure. The systolic growth of |$(H,|\cdot|)$| is the function mapping |$r\ge 0$| to the infimum |$\sigma(r)\in R_+\cup\{+\infty\}$| of covolumes of cocompact lattices of |$H$| with systole |$\ge r$|⁠. See Remark 3.1 for the geometric interpretation in a Riemannian setting. The definition makes sense when |$H$| is discrete, in which case lattices just refer to finite index subgroups: this is actually the setting in Gromov’s original definition. In the setting we will focus on, |$H$| will always be nilpotent and in this case all lattices are cocompact. In general, we can define another type of systolic growth, allowing non-cocompact lattices. The asymptotics of the growth of |$\sigma$| does not depend on the choice of the word length. The number |$\sigma(r)$| is always bounded below by the volume of the open |$r/2$|-ball in |$H$|⁠. The function |$\sigma$| is interesting only when it takes finite values, in which case we say that |$H$| is residually systolic. When |$H$| is discrete, this just means that |$H$| is residually finite. In general, a sufficient condition for |$H$| being residually systolic is that |$H$| admits a residually finite cocompact lattice. It is natural to compare the volume growth and the systolic growth. For finitely generated linear groups of exponential growth, the systolic growth is exponential as well [5]. 1.2 Background with focus in the nilpotent case Given a Lie algebra |${\mathfrak{g}}$|⁠, denote by |$({\mathfrak{g}}^i)_{i\ge 1}$| its lower central series (see Section 2.1); by definition |${\mathfrak{g}}$| is |$c$|-step nilpotent if |${\mathfrak{g}}^{c+1}=\{0\}$|⁠. The homogeneous dimension of |${\mathfrak{g}}$| is classically defined as the sum \[D=D({\mathfrak{g}})=\sum_{i\ge 1}\dim({\mathfrak{g}}^i);\] we have |$D<\infty$| if and only |${\mathfrak{g}}$| is nilpotent and finite-dimensional. A classical result of Malcev is that a simply connected nilpotent Lie group admits a lattice (which is then cocompact and residually finite) precisely when its Lie algebra can be obtained from a rational Lie algebra by extension of scalars. Therefore this is also equivalent to being residually systolic. In this case, the systolic growth is easy to bound polynomially; nevertheless the comparison between volume growth and the systolic growth is not obvious, because the precise rate of polynomial growth of the latter is an issue. A first step towards a good understanding is the following result (all asymptotic results are meant when |$r\to +\infty$|⁠). Theorem 1.2 ([9]). Let |$G$| be a simply connected nilpotent Lie group with a lattice |$\Gamma$|⁠. Let |${\mathfrak{g}}$| be the Lie algebra of |$G$|⁠, and let |$D$| be its homogeneous dimension (so, for both |$G$| and |$\Gamma$|⁠, the growth is |$\simeq r\mapsto r^D$| and the systolic growth is |$\succeq r^D$|⁠, see Section 2.1). The following are equivalent: (1) the systolic growth of |$G$| is |$\simeq r^D$|⁠; (2) the systolic growth of |$\Gamma$| is |$\simeq r^D$|⁠; (3) |${\mathfrak{g}}$| is Carnot, that is, admits a Lie algebra grading |${\mathfrak{g}}=\bigoplus_{i\ge 1}{\mathfrak{g}}_i$| such that |${\mathfrak{g}}_1$| generates |${\mathfrak{g}}$|⁠. Otherwise, if |$\sigma$| denotes the systolic growth of either |$\Gamma$| or |$G$|⁠, then it satisfies |$\sigma(r)\gg r^D$|⁠. In the non-Carnot case, the proof of Theorem 1.2 does not provide any explicit asymptotic lower bound improving |$\sigma(r)\gg r^D$|⁠. In this article, we carry out the task of evaluating the systolic growth in a number of explicit non-Carnot cases. Such results are presented in Section 1.4. We start with some general upper bounds. We will often emphasize the quotient |$\sigma'(r)=r^{-D}\sigma(r)$|⁠, since it often naturally occurs in computations, and its growth is a measure of the failure of being Carnot. 1.3 Upper bounds We provide here some upper bounds on the systolic growth. We denote by |$\lceil\cdot\rceil$| the ceiling function. Given |$c\ge 0$|⁠, we define \[k_c({\mathfrak{g}})=\sum_{i=1}^{\lceil c/2\rceil-1}\left(\frac{c}2-i\right)\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1});\] Note that |$k_c({\mathfrak{g}})\le\left(\frac{c}2-1\right)\dim({\mathfrak{g}}/{\mathfrak{g}}^{\lceil c/2\rceil})$|⁠. Proposition 1.3 (See Proposition 5.1). Let |${\mathfrak{g}}$| be a finite-dimensional |$c$|-step nilpotent real Lie algebra with homogeneous dimension |$D$|⁠, and let |$k=k_c({\mathfrak{g}})$| be defined as above. Assume that the corresponding simply connected nilpotent Lie group |$G$| admits lattices. Then the systolic growth |$\sigma(r)$| of |$G$| and its lattices is |$\preceq r^{D+k}$|⁠. We have |$k_c({\mathfrak{g}})\le\frac16\dim({\mathfrak{g}})^2$| (see Proposition 5.2) when |$c$| is the nilpotency length of |${\mathfrak{g}}$|⁠, so that we obtain an upper bound on |$\sigma(r)/r^D$| depending only on |$\dim({\mathfrak{g}})$|⁠. For small values of |$c$|⁠, we have \begin{gather*} k_{\le 2}({\mathfrak{g}})=0;\quad k_3({\mathfrak{g}})=\frac12\dim({\mathfrak{g}}/{\mathfrak{g}}^2),\quad k_4({\mathfrak{g}})=\dim({\mathfrak{g}}/{\mathfrak{g}}^2);\\ k_5({\mathfrak{g}})=\frac32\dim({\mathfrak{g}}/{\mathfrak{g}}^2)+\frac12\dim({\mathfrak{g}}^2/{\mathfrak{g}}^3);\quad k_6({\mathfrak{g}})=2\dim({\mathfrak{g}}/{\mathfrak{g}}^2)+\dim({\mathfrak{g}}^2/{\mathfrak{g}}^3). \end{gather*} Also note that |$D({\mathfrak{g}})+k_c({\mathfrak{g}})\le c\dim({\mathfrak{g}})/2$|⁠. Proposition 1.3 improves the trivial upper bound |$\preceq r^{c\dim({\mathfrak{g}})}$|⁠, making use of congruence subgroups in a lattice, which was mentioned in [9]. Every 2-step nilpotent Lie algebra is Carnot; the smallest nilpotency length allowing non-Carnot Lie algebra is 3. When |${\mathfrak{g}}$| is 3-step nilpotent, the above proposition yields |$\sigma(r)\preceq r^{D+\dim({\mathfrak{g}}/{\mathfrak{g}}^2)/2}$|⁠. This bound is not very far from sharp, see Theorem 1.7. 1.4 Lower bounds on the systolic growth and precise estimates This part is the bulk of the article. It contains the first exact estimates of the asymptotic behavior of the systolic growth of nilpotent Lie groups beyond the Carnot case covered by Theorem 1.2. The first non-Carnot Lie algebras occur in dimension 5 and in this case we have the following theorem. Theorem 1.4. Let |$G$| be a five-dimensional simply connected nilpotent Lie group whose Lie algebra |${\mathfrak{g}}$| is non-Carnot (there are 2 non-isomorphic possibilities for |${\mathfrak{g}}$|⁠, for which the homogeneous dimension |$D$| is either 8 or 11). Then the systolic growth of both |$G$| and its lattices is |$\simeq r^{D+1}$|⁠. □ Both cases are obtained in a single proof. In dimension 6, the classification yields 13 non-Carnot nilpotent real Lie algebras; a similar approach provides precise estimates for at least some of them, but I do not know if it can exhibit a behavior different from being |$\simeq r^{D+h}$| with |$h\in\{1,2,3\}$|⁠. In dimension 7, where a classification is still known (but lengthy), a similar approach yields an example for which the degree is not an integer: Theorem 1.5. There exists a seven-dimensional simply connected nilpotent Lie group for which the systolic growth, as well as the systolic growth of one of its lattices, is |$\simeq r^{D+3/2}$|⁠. This contrasts with the fact that the volume growth always has an integral degree of polynomial growth (the homogeneous dimension |$D$|⁠). Yet so far we only know, for the systolic growth, behaviors of the form |$r^{D+h}$| with |$h$| a non-negative rational, but we actually do not know if |$\log\sigma(r)/\log(r)$| always converges, and if so, if its limit is always a rational, and what kind of further constraints we can expect on |$h$| (see the questions below). At the computational level, let us also provide some families of unbounded dimension, for which we obtain unbounded values for |$h$|⁠. Theorem 1.6 (Truncated Witt Lie algebra). For |$n\ge 3$|⁠, let |$G(n)$| be the simply connected nilpotent Lie group corresponding to the Lie algebra |${\mathfrak{g}}(n)$| with basis |$(e_i)_{1\le i\le n}$| and nonzero brackets |$[e_i,e_j]=(i-j)e_{i+j}$|⁠, (⁠|$i+j\le n$|⁠). Then its systolic growth grows as |$r\mapsto r^{D+h}$| with |$h=\lceil (n-4)/2\rceil$| (here the homogeneous dimension is |$D=\frac{n(n-1)}{2}+1$|⁠). The following family of examples with unbounded |$h$| consists of 3-step nilpotent Lie groups. Theorem 1.7. (see Theorem 7.6). For |$n\ge 0$|⁠, let |${\mathfrak{g}}(4+2n)$| be the 3-step nilpotent |$(4+2n)$|-dimensional Lie algebra obtained as central product of a four-dimensional filiform Lie algebra and a |$(2n+1)$|-dimensional Heisenberg Lie algebra. Let |$G(4+2n)$| be the corresponding simply connected nilpotent Lie group. Then its systolic growth grows as |$r\mapsto r^{D+n}$|⁠, where |$D=2n+7$| is the homogeneous dimension. The same method actually yields examples for which the polynomial degree of |$r^{-D}\sigma(r)$| is comparable to the square of the dimension, see Remark 7.8. 1.5 Outline of the method Let us outline the method used to obtained the estimates of Section 1.4. The first step is to translate the problem, which concerns lattices in a simply connected nilpotent Lie group, into a problem about discrete cocompact subrings in its Lie algebra. This uses the fact that even if there is no exact correspondence between the two (the exponential of a Lie subring can fail to be a subgroup, and the logarithm of a subgroup can fail to be a Lie subring), this is true “up to bounded index”. This important fact, for which we claim no originality but could not refer to a written proof, is formulated in Lemma 4.1 and proved (along with a more precise statement) in the Appendix. Then the notion of systolic growth can be made meaningful in a real finite-dimensional nilpotent Lie algebra: it maps |$r$| to the smallest covolume of a discrete cocompact subring of Guivarch systole |$\ge r$|⁠. Here the Guivarch systole is the Lie algebra counterpart of the systole: this is the smallest Guivarch length of a nontrivial element in the lattice. The Guivarch length is recalled in Section 2.1; for instance, in the three-dimensional Heisenberg Lie algebra, the Guivarch length of an element can be defined as the value |$|x|+|y|+|z|^{1/2}$|⁠. See Section 4.2. The previous fact shows that the systolic growth of a simply connected nilpotent Lie group is asymptotically equivalent to that of its Lie algebra. Next, we have to estimate the systolic growth in various Lie algebras. The idea is to use a flag of rational ideals |${\mathfrak{g}}=\mathfrak{w}_1\ge \dots \dots\mathfrak{w}_k=\{0\}$|⁠. Here, if we consider arbitrary lattices, we need these ideals to be rational for every rational structure (we call this solid and provide some basic facts about such ideals in Section 2.2). For instance, terms of the lower central series are solid ideals. Then any lattice intersects each |$\mathfrak{w}_i$| into a lattice and this intersection maps into a lattice in |$\mathfrak{w}_i/\mathfrak{w}_{i+1}$|⁠; let |$a_i$| be the corresponding covolume. Then the covolume of the whole lattice is |$\prod_ia_i$|⁠. Then we use the stability under brackets and the hypothesis of Guivarch systole |$\ge r$| to obtain lower bounds on |$\prod a_i$|⁠, which in some cases are better than the trivial lower bound (the trivial lower bound has the form |$\succeq r^D$|⁠). More precisely, this approach typically yields, for a lattice of Guivarch systole |$\ge r$|⁠, some inequalities of the form |$a_ia_j\ge a_k r^{m(i,j,k)}$| for some integer |$(i,j,k)$|⁠. If we write |$A_i=\log_r(a_i)$| (so that the covolume is |$r^{\sum A_i}$|⁠), this can be rewritten as |$A_i+A_j\ge A_k+m(i,j,k)$|⁠. Then such a family of inequalities can yield a lower bound of the form |$\sum A_i\ge q$| for some rational |$q$|⁠, and thus yielding a lower bound for the covolume |$\prod a_i \ge r^q$|⁠. Once such a method is checked to yield precise estimates in some cases, it is not a surprise to find that in well-chosen examples, it yields non-integral degrees, as in Theorem 1.5. Let us also mention that we actually renormalize the problem by a well-chosen family of linear automorphisms (Section 6.1), which yields lower bounds for |$r^{-D}\sigma(r)$| and simplifies the computations (for instance, it allows to treat simultaneously both examples of Theorem 1.4). The approach also provides a new, simpler proof of the implication (1)|$\Rightarrow$|(3) in Theorem 1.2, see Section 6.3. 1.6 Open questions Let us mention some open problems. Since the number |$h$| is possibly not always defined, we write things as follows. Let |$H$| be a compactly generated, locally compact group of polynomial growth, admitting at least a lattice (we especially have in mind the cases when |$H$| is a simply connected nilpotent Lie group, or |$H$| is a finitely generated nilpotent group). So its systolic growth |$r\mapsto\sigma(r)$| is well-defined. Let |$D$| be its homogeneous dimension. Define \[\underline{h_H}=\underline{\lim}\frac{\log(\sigma(n)/n^D)}{\log(n)};\quad\overline{h_H}=\overline{\lim}\frac{\log(\sigma(n)/n^D)}{\log(n)}\] Question 1.8. (1) Is it always true that |$\underline{h_H}=\overline{h_H}$|? (I conjecture a positive answer). (2) Are |$\underline{h_H}$| and |$\overline{h_H}$| always rational numbers? (3) Is it true that |$\sigma_H(n)\simeq n^{D+h'}$| for some |$h'\ge 0$|? (Of course this implies a positive answer to (1), but this is more optimistic and I do not conjecture anything). Question 1.9. Does there exist infinitely many non-equivalent types of systolic growth asymptotically bounded above by some given polynomial? Question 1.10. Let |$G$| be a simply connected nilpotent Lie group with a lattice |$\Gamma$|⁠, with systolic growth |$\sigma_G$| and |$\sigma_\Gamma$| (so |$\sigma_G\preceq\sigma_\Gamma$|⁠). (1) Do we always have |$\sigma_G\simeq\sigma_\Gamma$|? (2) We now refer to the uniform systolic growth introduced in Section 3.3. We have |$\sigma_G\preceq\sigma^u_G\preceq\sigma^u_{\Gamma,G}$|⁠. Do we always have |$\sigma_G\simeq\sigma^u_G$|? Do we always have |$\sigma^u_G\simeq\sigma^u_{\Gamma,G}$|? 2 Algebraic Preliminaries In all this section, |$K$| is a field of characteristic zero, unless explicitly specified. 2.1 Lie algebras: lower central series, growth Let |${\mathfrak{g}}$| be a nilpotent fd Lie algebra over |$K$| (fd stands for finite-dimensional). Its lower central series is defined by |${\mathfrak{g}}^1={\mathfrak{g}}$|⁠, |${\mathfrak{g}}^{i+1}=[{\mathfrak{g}},{\mathfrak{g}}^i]$| for |$i\ge 1$|⁠. Let |$\mathfrak{v}_i$| be a supplement subspace of |${\mathfrak{g}}^{i+1}$| in |${\mathfrak{g}}^i$|⁠, so that |${\mathfrak{g}}=\bigoplus_{i\ge 1}\mathfrak{v}_i$| (and |$\mathfrak{v}_i=0$| for large |$i$|⁠). We have, for all |$i,j$| \[ [\mathfrak{v}_i,\mathfrak{v}_j]\subset \bigoplus_{k\ge i+j}\mathfrak{v}_k,\qquad \mathfrak{v}_{i+1}\subset [\mathfrak{v}_1,\mathfrak{v}_i]+{\mathfrak{g}}^{i+2}.\] Let |$G$| be the group of |$K$|-points of the corresponding unipotent algebraic group; |$G$| can obtained from |${\mathfrak{g}}$| using the Baker–Campbell–Hausdorff formula as group law. When |$K=\mathbf{R}$|⁠, this is the simply connected Lie group associated to |${\mathfrak{g}}$|⁠. The integer \[D=D({\mathfrak{g}})=D(G)=\sum_i\dim(\mathfrak{v}_i)=\sum_i\dim{\mathfrak{g}}^i\] is called the homogeneous dimension of |$G$|⁠. Indeed, in the real case, the volume of the |$r$|-ball is |$\simeq r^D$| [13] (see Section 3.1 for the definition of |$\simeq$|⁠); this degree formula was also found by Bass [1] while restricting to the discrete setting. We have |$D\ge\dim({\mathfrak{g}})$|⁠, with equality if and only if |${\mathfrak{g}}$| is abelian. Again in the real case, fix a norm on each |$\mathfrak{v}_i$|⁠. If |$x=(x_1,x_2,\dots)\in{\mathfrak{g}}$| in the decomposition |${\mathfrak{g}}=\bigoplus_{i\ge 1}\mathfrak{v}_i$|⁠, define its Guivarch length |${\lfloor} x{\rfloor}=\sup_i\|x_i\|^{1/i}$|⁠. This length plays an important role, as Guivarch established that |${\lfloor} x{\rfloor}$| is a good estimate for the word length of |$\exp(x)$| in the simply connected nilpotent Lie group |$G$| associated to |${\mathfrak{g}}$|⁠. 2.2 Solid ideals We introduce here the notion of solid ideals, which will be useful when computing lower bounds on the systolic growth of simply connected nilpotent Lie groups. Let |${\mathfrak{g}}$| be a Lie algebra (over |$K$|⁠). A |$\mathbf{Q}$|-structure on |${\mathfrak{g}}$| is the data of a |$\mathbf{Q}$|-subspace |$\mathfrak{l}$| such that the canonical homomorphism |$j:\mathfrak{l}\otimes_\mathbf{Q} K\to{\mathfrak{g}}$| is a linear isomorphism of Lie |$K$|-algebras. If in addition |$\mathfrak{l}$| is a |$\mathbf{Q}$|-subalgebra, we call it a multiplicative |$\mathbf{Q}$|-structure, and then |$j$| is a |$K$|-algebra isomorphism. Given a |$\mathbf{Q}$|-structure |$\mathfrak{l}$|⁠, a |$K$|-subspace |$V$| of |${\mathfrak{g}}$| is called |$\mathbf{Q}$|-defined if it is generated as a |$K$|-subspace by |$V\cap\mathfrak{l}$|⁠. If |${\mathfrak{g}}$| is a Lie |$K$|-algebra, we say that an ideal is solid if it is |$\mathbf{Q}$|-defined for every multiplicative |$\mathbf{Q}$|-structure. The following properties are straightforward. A ideal contained and solid in a solid ideal is solid in the whole Lie algebra; the inverse image of a solid ideal by the quotient by a solid ideal is solid; the bracket of two solid ideals is solid: the centralizer of a solid ideal is solid; the intersection and the sum of two solid subalgebras are solid. For instance, if |${\mathfrak{g}}$| is abelian, then the only solid ideals of |${\mathfrak{g}}$| are |$\{0\}$| and |${\mathfrak{g}}$|⁠. To single out solid ideals in fd |$\mathbf{Q}$|-definable real nilpotent Lie algebras |${\mathfrak{g}}$| is useful in view of the following: an ideal |$V\subset{\mathfrak{g}}$| is solid if and only if for every discrete cocompact subring |$\Lambda$|⁠, the intersection |$\Lambda\cap V$| is a lattice in |$V$| (or equivalently, the projection on |${\mathfrak{g}}/V$| is a lattice in |${\mathfrak{g}}/V$|⁠). The reason is that the |$\mathbf{Q}$|-linear span of any discrete cocompact subring is a multiplicative |$\mathbf{Q}$|-structure, and that conversely any multiplicative |$\mathbf{Q}$|-structure contains a discrete cocompact subring. Solid ideals can almost be recognized using how they behave under the automorphism group. Say that an ideal |$I$| in |${\mathfrak{g}}$| is absolutely |${\text{Aut}}$|-invariant if, denoting by |$\bar{K}$| an algebraic closure of |$K$| and |${\text{Aut}}({\mathfrak{g}})_{\bar{K}}$| for the group of automorphisms of the |$\bar{K}$|-algebra |${\mathfrak{g}}\otimes_K\bar{K}$|⁠, the ideal |$I\otimes_K\bar{K}$| is |${\text{Aut}}({\mathfrak{g}})_{\bar{K}}$|-invariant. Theorem 2.1. Assume that |$K$| is uncountable. Let |${\mathfrak{g}}$| be a fd Lie algebra over |$K$|⁠. Then (1) every solid ideal is invariant under |${\text{Aut}}({\mathfrak{g}})^0$| (or equivalently, stable under all |$K$|-linear self-derivations of |${\mathfrak{g}}$|⁠); (2) every absolutely |${\text{Aut}}$|-invariant ideal is solid. Remark 2.2. If |$I$| is an ideal of |${\mathfrak{g}}$|⁠, we have: |$I$| absolutely Aut-invariant |$\Rightarrow$||$I$||${\text{Aut}}({\mathfrak{g}})$|-invariant |$\Rightarrow$||$I$||${\text{Aut}}({\mathfrak{g}})^0$|-invariant (⁠|$H^0$| denoting the connected component of the unit in the Zariski topology). The reverse implications do not hold in general. For instance, in |$\mathfrak{sl}_2(K)\times\mathfrak{sl}_2(K)$|⁠, the ideal |$\mathfrak{sl}_2(K)\times\{0\}$| only satisfies the third condition. Also, over the reals, in |$\mathfrak{sl}_2(\mathbf{R})\times\mathfrak{so}_3(\mathbf{R})$|⁠, the ideal |$\mathfrak{sl}_2(\mathbf{R})\times\{0\}$| only satisfies the latter two conditions (however, it is solid). Over the complex numbers, I do not know whether every solid ideal is |${\text{Aut}}({\mathfrak{g}})$|-invariant. Theorem 2.1 follows from the next two propositions. Since the Lie algebras axioms plays no role here, we consider arbitrary algebras and the context could be even more general. Also, solid can be defined for arbitrary subspaces and we use this straightforward generalization. However, in a Lie algebra solid subspaces are always ideals (Corollary 2.6) Proposition 2.3. Let |${\mathfrak{g}}$| be a fd |$K$|-algebra and |$I$| a |$K$|-subspace, |$\mathbf{Q}$|-defined for at least one multiplicative |$\mathbf{Q}$|-structure; let |$\bar{K}$| be an algebraically closed extension of |$K$|⁠. If |$I\otimes_K\bar{K}$| is |${\text{Aut}}({\mathfrak{g}})_{\bar{K}}$|-invariant and |$\mathbf{Q}$|-defined for some multiplicative |$\mathbf{Q}$|-structure on the algebra |${\mathfrak{g}}$|⁠, then |$I$| is solid. Proof. Let |$\Xi$| be the Galois group of |$\bar{K}$| over |$\mathbf{Q}$|⁠. Let |$V$| be a finite-dimensional |$K$|-vector space. Let |$W\subset V$| be a |$\mathbf{Q}$|-structure in |$V$|⁠. Then |$\Xi$| acts coordinate-wise on |$V_{\bar{K}}=V\otimes_K\bar{K}=W\otimes_\mathbf{Q}\bar{K}$| (for some choice of basis of |$W$|⁠, whose choice does not matter). We denote this action as |$\gamma\cdot v=u_W(\gamma)v$| for |$\gamma\in\Xi$|⁠. Then |$u_W(\gamma)$| is a |$\mathbf{Q}$|-linear automorphism of |$V_{\bar{K}}$|⁠; it is also |$\gamma$|-semi-linear, in the sense that |$u_W(\gamma)(\lambda v)=\gamma(\lambda)u_W(\gamma)v$|⁠. It follows in particular that if |$W'$| is another |$\mathbf{Q}$|-structure, then |$\eta_{W,W'}(\gamma)=u_W^{-1}(\gamma)u_{W'}(\gamma)$| is a |$\bar{K}$|-linear automorphism of |$V_{\bar{K}}$|⁠. Now suppose that the action |$u_W$| of |$\Xi$| leaves a |$\bar{K}$|-subspace |$J\subset V_{\bar{K}}$| invariant. This implies that |$J$| is the |$\bar{K}$|-linear span of |$J\cap W$| (see [4, corollaire in Chapter V.4]). We apply this to |$J=I_{\bar{K}}=I\otimes_K\bar{K}$|⁠. By assumption, |$I$| is |$\mathbf{Q}$|-defined for some multiplicative |$\mathbf{Q}$|-structure |$W'$|⁠. So |$J$| is |$u_{W'}(\Xi)$|-invariant. It is also |${\text{Aut}}({\mathfrak{g}})(\bar{K})$|-invariant, by assumption. Hence, by the first paragraph of the proof, it is |$u_W(\Xi)$|-invariant. So |$I_{\bar{K}}$| is the |$\bar{K}$|-linear span of |$J\cap W$|⁠. This means that |$\dim_\mathbf{Q}(J\cap W)=\dim_{\bar{K}}(I_{\bar{K}})$|⁠. Since the latter equals |$\dim_K(I)$| and since |$J\cap W=I\cap W$|⁠, this in turn implies that |$I$| is the |$K$|-linear span of |$I\cap W$|⁠, that is, |$I$| is |$\mathbf{Q}$|-defined. ■ Proposition 2.4. Let |${\mathfrak{g}}$| be a fd algebra over an uncountable field of characteristic zero, definable over |$\mathbf{Q}$|⁠. Let |$V$| be a subspace of |${\mathfrak{g}}$|⁠. Suppose that |$V$| is solid, that is, it is |$\mathbf{Q}$|-defined for every |$\mathbf{Q}$|-structure on |${\mathfrak{g}}$|⁠. Then |$V$| is invariant under |$H={\text{Aut}}({\mathfrak{g}})^0$|⁠, that is, when |$K$| has characteristic zero, |$V$| is stable under every derivation of |${\mathfrak{g}}$|⁠. Proof. Let |$L\subset H$| be the stabilizer of |$V$|⁠. If |$L\neq H$|⁠, then |$H(K)/L(K)$| is uncountable: if |$K$| is |$\mathbf{R}$| or |$\mathbf{C}$| this is because it is a manifold of positive dimension; in general see Lemma 2.5. The map |$hL\to hV$| from |$H(K)/L(K)$| to the set of subspaces of |${\mathfrak{g}}$| being injective, it has an uncountable image. So, given a |$\mathbf{Q}$|-structure |${\mathfrak{g}}_\mathbf{Q}$|⁠, there exists |$h\in H(K)$| such that |$hV$| is not |$\mathbf{Q}$|-defined. Accordingly, for the new |$\mathbf{Q}$|-structure defined by |$h^{-1}{\mathfrak{g}}_\mathbf{Q}$|⁠, |$V$| is not |$\mathbf{Q}$|-defined, contradicting that |$V$| is solid. ■ Lemma 2.5. Let |$H$| be a connected linear algebraic group defined over an infinite perfect field |$K$|⁠, and |$L$| a |$K$|-closed proper subgroup. Then |$H(K)/L(K)$| is has the same cardinality as |$K$|⁠. Proof. (Beware that the canonical injective map |$H(K)/L(K)\to (H/L)(K)$| can fail to be surjective, so it is not enough to compute the cardinal of the latter). Clearly the cardinal of |$H(K)$| is bounded above by that of |$K$| (as soon as |$K$| is infinite), so we have to prove the other inequality. We first take for granted that there exists a |$K$|-closed curve |$C$| in |$H$|⁠, |$K$|-birational to |$\mathbb{P}^1$|⁠, such that |$C\nsubseteq L$| and |$1\in C$|⁠. This granted, let us conclude (without the perfectness restriction on |$K$|⁠). On |$C$|⁠, we consider the equivalence relation |$x\sim y$| if |$x^{-1}y\in L$|⁠. This is a closed subvariety of |$C\times C$|⁠, and does not contain any layer |$C\times\{y_0\}$| since |$y_0\in L$| and then |$C\subset L$| would follow. So equivalence classes are finite. Since |$C$| is |$K$|-birational to |$\mathbb{P}^1$|⁠, its cardinal is the same as |$K$|⁠, and then its image in the quotient |$H(K)/L(K)$| being the quotient of |$C(K)$| by the equivalence relation |$\sim$| with finite classes, it also has at least the cardinal of |$K$|⁠. To justify the existence of |$C$|⁠, we can argue that |$H$| is |$K$|-unirational (this uses that |$K$| is perfect), as established in [2, Corollary 7.12]; consider a dominant |$K$|-defined morphism |$f:U\to H$|⁠, with |$U$| open in the affine |$d$|-space |$\mathbb{A}^d$|⁠. Conjugating with translations both in |$\mathbb{A}^d$| and |$H$|⁠, we can suppose that |$0\in U$| and |$f(0)=1$|⁠. Since |$U(K)$| is Zariski-dense and |$f$| is dominant, |$f(U)$| is Zariski-dense, and hence it contains a point |$f(x)\notin L$| for some |$x\in U$|⁠. Let |$D\subset\mathbb{A}^d$| be the line through |$x$|⁠. Then the Zariski-closure of |$f(D\cap U)$| is the desired curve. ■ Corollary 2.6. In a fd Lie algebra over a field of characteristic 0, every solid subspace |$V$| is an ideal. Proof. We have to show that if |$V$| is |${\text{Aut}}({\mathfrak{g}})_0$|-invariant then it is an ideal. Indeed, differentiation implies that |$V$| is invariant under derivations of |${\mathfrak{g}}$|⁠, and this includes inner derivations. Since left and right multiplications are derivations, this finishes the proof. ■ Example 2.7. (A solid complete flag in a filiform Lie algebra of dimension |$\ge 4$|⁠). Recall that, for |$d\ge 2$|⁠, a filiform Lie algebra denotes a |$d$|-dimensional nilpotent Lie algebra whose nilpotency length is exactly |$d-1$|⁠, which is the largest possible value. Such a Lie algebra |${\mathfrak{g}}$| admits a basis |$(e_1,\dots,e_d)$| such that, denoting by |${\mathfrak{g}}_{\ge i}$| the subspace with basis |$(e_i,\dots,e_d)$|⁠, we have, for all |$i\in\{2,\dots,d\}$|⁠, |${\mathfrak{g}}^i={\mathfrak{g}}_{\ge i+1}$|⁠. In addition, if |$d\ge 4$|⁠, we can arrange to choose |$[e_1,e_2]=e_3$| and |$[e_1,e_3]=e_4$|⁠. This being assumed, each |${\mathfrak{g}}_{\ge i}$| is a solid ideal. Indeed, for |$i\neq 2$|⁠, this is because it is a term of the lower central series. For |$i=2$|⁠, this is because it is the centralizer of |${\mathfrak{g}}^2(={\mathfrak{g}}^{\ge 3})$| modulo |${\mathfrak{g}}^4(={\mathfrak{g}}_{\ge 5})$|⁠. 3 Systolic Growths: Facts and Bounds 3.1 Asymptotic comparison Given functions |$f,g$| (of a positive real variable |$r$|⁠), we write |$f\preceq g$| if |$f(r)\le Cg(C'r)+C''$| for some constants |$C,C',C''>0$| and all |$r$|⁠. We say that |$f$| and |$g$| are |$\simeq$|-equivalent and write |$f\simeq g$| if |$f\preceq g\preceq f$|⁠. Also, we write |$f\ll g$| if |$g/(|f|+1)\to +\infty$| (usually |$\underline{\lim}_{r\to\infty}f>0$|⁠, in which case this just means |$g/f\to +\infty$|⁠). 3.2 Residual girth The residual girth is defined in the same way as the systolic growth, but allowing only normal subgroups. Let |$G$| be a simply connected nilpotent Lie group, of dimension |$d$| and nilpotency length |$c$| with a lattice |$\Gamma$|⁠. We can find an embedding of |$G$| into the group of upper triangular unipotent real matrices, mapping |$\Gamma$| into integral matrices. Then congruence subgroups (the kernel of reduction modulo |$n$|⁠, in restriction to |$\Gamma$|⁠) have index |$\simeq n^d$| and systole |$\simeq n^{1/c}$|⁠. This yields for |$\Gamma$| the polynomial upper bound on the residual girth |$\preceq r\mapsto r^{cd}$|⁠; this simple observation was made independently in [6, 9]. As observed in [9], in the case of the three-dimensional Heisenberg group, it is sharp: the residual girth of every lattice is |$\simeq r^6$|⁠. In [6], it is shown that, more generally, the residual girth of every lattice is indeed |$\simeq r^{cd}$| when the center of |$G$| coincides with the |$c$|-th term of the central series. Otherwise the picture is not completely clear. The above provides an easy polynomial upper bound on the systolic growth of |$\Gamma$| and of |$G$| which are also |$\preceq r^{cd}$|⁠. Nevertheless, we will not concentrate further on the residual girth |$\sigma_\Gamma^\lhd$|⁠, inasmuch as its asymptotic behavior is usually much larger than the systolic growth |$\sigma_\Gamma$|⁠: as soon as |$G$| is non-abelian, |$\sigma_\Gamma(r)\ll r^{cd}$|⁠, and it is also very likely that |$\sigma_\Gamma\ll\sigma_\Gamma^\lhd$| always holds. 3.3 Systolic growths Another notion, also related with conjugacy phenomena, but more closely related to the systolic growth, is the uniform systolic growth introduced in [9]: we define it now. Let |$H$| be an arbitrary compactly generated locally compact group. We use the standard natural convention |$\inf\emptyset=+\infty$|⁠. In the setting we will study more deeply (compactly generated locally compact nilpotent groups), all lattices are cocompact. So we stick to cocompact lattices, but in general it would make sense to consider the analogous notion allowing arbitrary lattices. In the setting of Definition 1.1, one can define the uniform systole, or |$H$|-uniform systole, of |$X\subset H$| as the infimum \[\inf_{h\in H}\mathrm{sys}(hXh^{-1})=\inf\{|hxh^{-1}|\}:\;h\in H,x\in X\smallsetminus\{1\}.\] The uniform (or |$H$|-uniform) systolic growth of |$H$| is then defined as the function mapping |$r>0$| to the infimum |$\sigma^u(r)\in\mathbf{R}_+\cup\{+\infty\}$| of covolumes of cocompact lattices of |$H$| with uniform systole |$\ge r$|⁠. Given a cocompact lattice |$\Gamma$| in |$H$|⁠, we can consider the uniform systolic growth |$\sigma_\Gamma$| of |$\Gamma$| (computed within |$\Gamma$|⁠) and the uniform systolic growth |$\sigma_H$| of |$H$|⁠. But while |$\sigma_H\preceq\sigma_\Gamma$|⁠, a similar estimate for the uniform systolic growths might fail because for a finite index subgroup of |$\Gamma$|⁠, the |$H$|-uniform systole can be much smaller than the |$\Gamma$|-uniform systole (see Example 3.2, in the Heisenberg group). Hence another notion naturally appears: the |$H$|-uniform systolic growth |$\sigma^u_{\Gamma,H}$| of |$\Gamma$|⁠, considering the uniform systole computed in |$H$| but finite index subgroups of |$\Gamma$|⁠. At this point we have a bunch of growths, which we summarize now (up to asymptotic equivalence, allowing to not specify the choice of lengths): each maps |$r\ge 0$| to the smallest covolume of a cocompact lattice of |$H$| with the additional conditions: (systolic growth |$\sigma_H$| of |$H$|⁠): of systole |$\ge r$|⁠; (systolic growth |$\sigma_\Gamma$| of |$\Gamma$|⁠: contained in |$\Gamma$|⁠, of systole |$\ge r$|⁠; (uniform systolic growth |$\sigma^u_H$| of |$H$|⁠): of |$H$|-uniform systole |$\ge r$| (uniform systolic growth |$\sigma^u_\Gamma$| of |$\Gamma$|⁠): contained in |$\Gamma$|⁠, of |$\Gamma$|-uniform systole |$\ge r$| (⁠|$H$|-uniform systolic growth |$\sigma^u_{\Gamma,H}$| of |$\Gamma$|⁠): contained in |$\Gamma$|⁠, of |$H$|-uniform systole |$\ge r$|⁠. with asymptotic inequalities \[ \sigma_H\preceq\sigma_\Gamma\preceq\sigma_\Gamma^u\preceq\sigma_{\Gamma,H}^u;\quad \sigma_H\preceq\sigma_H^u\preceq\sigma^u_{\Gamma,H}. \] (Let us also mention that all these functions are |$\preceq\sigma^\lhd_\Gamma$|⁠: the only nontrivial case is that of |$\sigma^u_{\Gamma,H}$|⁠, and follows from the fact that for a subgroup |$\Lambda$| normalized by a fixed cocompact lattice |$\Gamma$|⁠, the |$H$|-normal systole is bounded by the |$\Gamma$|-uniform systole plus a constant depending only on |$\Gamma$|⁠, independently of |$\Lambda$|⁠). In the context when |$H=G$| is a simply connected nilpotent Lie group, all these functions take finite values and are asymptotically bounded above by the residual girth of |$\Gamma$|⁠, and in particular are polynomially bounded by |$r\mapsto r^{c\dim(G)}$|⁠. A better upper bound (implying |$\preceq r^{c\dim(G)/2}$|⁠) is provided in Proposition 5.1. In this context, we do not know if all these five functions have the same asymptotic behavior (Question 1.10). In various cases where we prove an upper bound on the systolic growth, by constructing an explicit sequence of lattices, we actually provide an upper bound on |$\sigma_{\Gamma,G}^u$| and therefore on all others. Remark 3.1. Most of these functions can be interpreted in the geometry of |$G/\Gamma$|⁠. More precisely, endow |$G$| with a right-invariant Riemannian metric, which thus passes to the quotient |$G/\Gamma$|⁠, as well as its covering |$G/\Lambda$| when |$\Lambda$| is a finite index subgroup of |$\Gamma$|⁠. Then |$\Gamma$| can naturally be identified to |$\pi_1(G/\Gamma)$| by a bijection |$\gamma\mapsto j_\gamma$| (the base-point is meant to be the obvious one). The length of |$\gamma\in\Gamma$| is equivalent to the length in |$X=G/\Gamma$| of a smallest representative based loop of |$j_\gamma$|⁠. Its |$G$|-uniform length is equivalent to the infimum of lengths of arbitrary loops in the free homotopy class of |$j_\gamma$|⁠. Therefore, the systole (resp. |$G$|-uniform systole) of |$\Lambda$| is equivalent the smallest size of a based loop (resp. of a loop) in |$X$| not homotopic to a point. Call the latter the geometric based systole, resp. geometric systole, of |$X$| (classically, the word “geometric” is dropped, since these notions come from Riemannian geometry!). The geometric based systolic growth, resp. geometric systolic growth, of |$X$| is defined as the function mapping |$r$| to the smallest degree of a covering of |$X$| with geometric based systole, resp. systole |$\ge r$|⁠. Thus the geometric based systolic growth of |$X$| coincides with the systolic growth of |$\Gamma$|⁠, and the geometric systolic growth of |$X$| coincides with the |$G$|-uniform systolic growth of |$\Gamma$|⁠. Example 3.2. In the three-dimensional real Heisenberg group |$G$|⁠, let us write, for the sake of shortness, . We write |$|M(a,b,c)|=|a|+|b|+\sqrt{|c|}$|⁠; we use this approximation of the distance to the origin to compute systoles. Let |$\Gamma_n$| be the subgroup generated by the matrices |$x_n=M(1,0,n)$|⁠, |$y_n=M(0,n^2,0)$|⁠; this is a lattice. We claim that its |$G$|-uniform systole is 1 while its |$\Gamma_n$|-uniform systole is |$\ge\sqrt{n}$|⁠. Let us describe |$\Gamma_n$|⁠. Define |$z_n=x_ny_nx_n^{-1}y_n^{-1}$|⁠. Then |$z_n=M(0,0,n^2)$| and elements of |$\Gamma_n$| are precisely those |$x_n^ay_n^bz_n^c$| when |$(a,b,c)$| ranges over |$\mathbf{Z}^3$|⁠. We see that \[x_n^ay_n^bz_n^c=M(a,bn^2,n(a+(ab+c)n)).\] In |$G$|⁠, |$x_n$| is conjugate to |$M(1,0,t)$| for every real |$t$|⁠, and hence the |$G$|-systole of |$\Gamma_n$| is equal to 1. Let us compute the |$\Gamma_n$|-uniform systole. Consider |$(a,b,c)\in\mathbf{Z}^3\smallsetminus\{0\}$| and consider |$g=x_n^ay_n^bz_n^c$|⁠, and |$g'$| any conjugate of |$g$| (for the moment, a |$G$|-conjugate). If |$(a,b)=(0,0)$|⁠, then |$c\neq 0$| and |$g=M(0,0,cn^2)$| is central, so |$|g'|=|g|=\sqrt{|c|}n\ge n$|⁠; if |$b\neq 0$|⁠, then |$|g'|\ge |a|+|b|n^2\ge n^2$|⁠; if |$b=0$| and |$a\neq 0$|⁠, then |$g=x_n^az_n^c=M(a,0,n(a+cn))$|⁠. We discuss if |$|a|\ge\sqrt{n}$|⁠, then |$|g'|\ge\sqrt{n}$|⁠; if |$|a|<\sqrt{n}$|⁠, we now assume that |$g'$| is a |$\Gamma_n$|-conjugate of |$g$|⁠, namely by some element |$M(*,dn^2,*')$| with |$d\in\mathbf{Z}$| (we do not have to care about the coefficients denoted by stars); this gives \[g'=M(a,0,n(a-(ad-c)n)).\] If by contradiction |$|g'|<\sqrt{n}$|⁠, then |$a-(ad-c)n=0$|⁠, but since |$|a|<n$| this implies |$a=0$|⁠, a contradiction. So |$|g'|\ge\sqrt{n}$|⁠. We conclude that in all cases |$|g'|\ge\sqrt{n}$|⁠, so the |$\Gamma_n$|-systole of |$\Gamma_n$| is |$\ge\sqrt{n}$|⁠. □ 4 Algebraization of the Systolic Growth The purpose of this section is to describe the various systolic notions in terms of discrete cocompact subrings of the Lie algebra, instead of lattices in the Lie group. While the exponential of a discrete cocompact subring can fail to be a subgroup and the logarithm of a lattice can fail to be an additive subgroup or fail to be stable under taking brackets, the correspondence is true up to bounded index. This is the contents of the following lemma. Lemma 4.1 (Folklore). Let |${\mathfrak{g}}$| be a real fd nilpotent Lie algebra and |$G$| the corresponding simply connected nilpotent Lie group. There exists |$C\ge 1$| depending only on |$\dim({\mathfrak{g}})$| such that: (1) for every cocompact discrete subring |$\Lambda$| in |${\mathfrak{g}}$|⁠, there exist lattices |$\Gamma_1,\Gamma_2$| in |$G$| with |$\Gamma_1\subset\exp(\Lambda)\subset\Gamma_2$| and |$[\Gamma_2:\Gamma_1]\le C$|⁠. (2) for every lattice |$\Gamma$| in |$G$|⁠, there exist cocompact discrete subrings |$\Lambda_1,\Lambda_2$| in |${\mathfrak{g}}$| with |$\Lambda_1\subset\log(\Gamma)\subset\Lambda_2$| and |$[\Lambda_2:\Lambda_1]\le C$|⁠. We include a proof (of a slightly stronger statement) in the Appendix. 4.1 Systolic growth of a real nilpotent Lie algebra We define a Lie algebra analogue of the systole and systolic growth as follows (fix some Lebesgue measure on the vector space |${\mathfrak{g}}$|⁠). Recall that |${\lfloor}\cdot{\rfloor}$| denotes the Guivarch length, introduced in Section 2.1. Definition 4.2. If |$H\subset{\mathfrak{g}}$|⁠, define \[{\mathrm{sys}}(H)=\inf\{{\lfloor} h{\rfloor}:h\in H\smallsetminus\{0\}\}\in\mathbf{R}_+\cup\{+\infty\}.\] Also define the uniform (or |$G$|-uniform) systole as |${\mathrm{sys}}^u(H)=\inf_{g\in G}\mathrm{sys}(g\cdot H)$| (where |$G$| acts through the adjoint representation). The systolic (resp. uniform systolic) growth of |$({\mathfrak{g}},{\lfloor}\cdot{\rfloor})$| is the function mapping |$r\ge 0$| to the infimum |$\sigma(r)\in\mathbf{R}_+\cup\{+\infty\}$| of covolumes of cocompact discrete subrings of |${\mathfrak{g}}$| with systole (resp. uniform systole) |$\ge r$|⁠. The systolic growth also depends on the choice of normalization of the Lebesgue measure. Its asymptotic growth, however, only depends on the real Lie algebra |${\mathfrak{g}}$|⁠. Its interest lies in the following fact: Proposition 4.3. The systolic (resp. uniform systolic) growth of |${\mathfrak{g}}$| (as a Lie algebra) and the systolic (resp. uniform systolic) growth of |$G$| are |$\simeq$|-equivalent. Proof. The exponential map preserves the systole of subsets up to a bounded multiplicative error, by Guivarch’s estimates. So the proposition would be already proved if the exponential map were giving an exact correspondence between cocompact discrete subrings of |${\mathfrak{g}}$| and lattices in |$G$|⁠. This is not the case, but is, however, “true up to a bounded index error”, as explained in Lemma 4.1, which entails the result for systolic growth. The uniform case follows, using in addition that the exponential map |${\mathfrak{g}}\to G$| is |$G$|-equivariant, for the adjoint action on |${\mathfrak{g}}$| and the conjugation action on |$G$|⁠. ■ 4.2 Systolic growth of a rational nilpotent Lie algebra To estimate the systolic growth in the lattice, we need a similar notion pertaining to rational Lie algebras. Namely, let |$\mathfrak{l}$| be a fd nilpotent Lie algebra over |$\mathbf{Q}$|⁠. Define its “realification” |${\mathfrak{g}}=\mathfrak{l}\otimes_\mathbf{Q}\mathbf{R}$|⁠. Choose |$\mathfrak{v}_i$| and norms as above on |${\mathfrak{g}}$|⁠, so as to define the Guivarch length |${\lfloor}\cdot{\rfloor}$|⁠; so we have a notion of systole for subsets |${\mathfrak{g}}$|⁠. Also choose a Lebesgue measure on |${\mathfrak{g}}$|⁠. Definition 4.4. The systolic (resp. |$G$|-uniform systolic) growth of the rational Lie algebra |$(\mathfrak{l},{\lfloor}\cdot{\rfloor})$| is the function mapping |$r\ge 0$| to the infimum |$\sigma(r)\in\mathbf{R}_+\cup\{+\infty\}$| of covolumes of cocompact discrete subrings of |${\mathfrak{g}}$| contained in |$\mathfrak{l}$|⁠, with systole (resp. |$G$|-uniform systole) |$\ge r$|⁠. The only difference in the last definition is that we restrict to those subrings contained in |$\mathfrak{l}$|⁠. In particular, this function is bounded below by the systolic growth of |$G$| (relative to the same choice of norms, etc.). Also note that we did not attempt to define an analogue of the |$\Gamma$|-uniform systolic growth. Denoting |$L=\exp(\mathfrak{l})$| (which can be thought as the group |$G_\mathbf{Q}$| of |$\mathbf{Q}$|-points of |$G$| for a suitable rational structure), we have Proposition 4.5. If |$\Gamma$| is any lattice in |$G$| contained in |$L$|⁠, then the systolic (resp. |$G$|-uniform systolic) growth of |$\Gamma$| is |$\simeq$|-equivalent to the systolic (resp. |$G$|-uniform systolic) growth of the rational Lie algebra |$\mathfrak{l}$|⁠. Proof. An observation is that in Lemma 4.1, if |$\Lambda\subset\mathfrak{l}$| then automatically |$\Gamma_i\subset L$|⁠, and in the other direction if |$\Gamma\subset L$| then automatically |$\Lambda_i\subset\mathfrak{l}$|⁠. This being granted, the proof follows the same lines as that of Proposition 4.3. ■ 5 General Upper Bounds on the Systolic Growth We prove Proposition 1.3, giving here a more general result since we consider the uniform systolic growth. Proposition 5.1. Let |$G$| be a simply connected nilpotent Lie group with a lattice |$\Gamma$|⁠. Let |$D$| be its homogeneous dimension and let |$k$| be defined in Section 1.3. Then |$\sigma_{\Gamma,G}^u(r)\preceq r^{D+k}$|⁠, and hence the systolic growth and |$G$|-uniform systolic growth of both |$G$| and |$\Gamma$| are all |$\preceq r^{D+k}$|⁠. Proof. It is enough to show that the |$G$|-uniform systolic growth of |$\Gamma$| is |$\preceq r^{k'}$|⁠, where \[k'=\frac{c}2\dim({\mathfrak{g}}/{\mathfrak{g}}^{\lceil c/2\rceil})+\sum_{i=\lceil c/2\rceil}^c i\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1})=\sum_{1}^c \max(c/2,i)\dim\left({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1}\right).\] Let |$G_\mathbf{Q}\subset G$| be the rational Malcev closure of |$\Gamma$| and |${\mathfrak{g}}_\mathbf{Q}$| its Lie algebra. From the lower central series we choose supplement subspace to obtain a vector space decomposition |${\mathfrak{g}}=\bigoplus_{i=1}^c{\mathfrak{g}}_i$|⁠, with |$\bigoplus_{j\ge i}{\mathfrak{g}}_j={\mathfrak{g}}^i$| for all |$i$| (this is not necessarily a Lie algebra grading). We choose bases of these subspaces so as to ensure that all structure constants are integral; thus |${\mathfrak{g}}_i(\mathbf{Z})$| means the discrete subgroup generated by the given basis of |${\mathfrak{g}}_i$|⁠. We now define, for every square integer |$n$|⁠, |$\Lambda_n=\bigoplus_{i=1}^cn^{\max(c/2,i)}{\mathfrak{g}}_i(\mathbf{Z})$|⁠. We have to check that this is a subalgebra, namely that \[B_{ij}:=[n^{\max(c/2,i)}{\mathfrak{g}}_i(\mathbf{Z}),n^{\max(c/2,j)}{\mathfrak{g}}_j(\mathbf{Z})]\subset\Lambda_n\] for all |$i,j$|⁠. Indeed, keeping in mind that |$n$| is a square, \[B_{ij}\subset n^c[{\mathfrak{g}}_i(\mathbf{Z}),{\mathfrak{g}}_j(\mathbf{Z})]\subset n^c\bigoplus_{p=i+j}^c{\mathfrak{g}}_p(\mathbf{Z})\subset \Lambda_n.\] The |$G$|-uniform systole of |$\Lambda_n$| is |$\succeq n$|⁠: indeed, any nonzero element has the form |$w=\sum_{j\ge i}n^jv_j$| with |$v_i\in{\mathfrak{g}}_i(\mathbf{Z})\smallsetminus\{0\}$|⁠. So any |$G$|-conjugate of |$w$| has the form |$n^iv_i+\mu$| with |$\mu\in{\mathfrak{g}}^{i+1}$| and hence has norm |$\ge n^i$| (for the |$\ell^1$|-norm with respect to the fixed basis), and thus has Guivarch length |$\ge n$|⁠. Then |$\Lambda_n$| precisely has covolume |$n^{k'}$|⁠. Using that the exponential map |${\mathfrak{g}}\to G$| is |$G$|-equivariant and in view of Lemma 4.1, we deduce the desired upper bound. ■ The following proposition provides upper bounds on |$k$| and |$k'=k+D$| depending only on the dimension |$d$|⁠. Recall that the maximal homogeneous dimension |$D$| for given dimension |$d\ge 2$| is equal to |$\frac{d(d-1)}2+1$| and is precisely attained for filiform Lie algebras, which are those of maximal nilpotency class (namely |$d-1$|⁠). Proposition 5.2. Let |${\mathfrak{g}}$| be a finite-dimensional nilpotent Lie algebra of dimension |$d$|⁠, nilpotency length |$c$|⁠, and homogeneous dimension |$D$|⁠, and |$k=k_c({\mathfrak{g}})$| is defined in Section 1.3. Then \[k\le \frac{d^2}6 -\frac{d}2+\frac{1}{2}\quad\text{and}\quad k+D\le \frac{5d^2-4d}8.\] Proof. We denote |$b=\lceil c/2\rceil$| and |$d=\dim({\mathfrak{g}})$|⁠. Then \begin{align*} k&= \sum_{i=1}^{b-1}\left(\frac{c}2-i\right)\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1}) \\ & = \frac{c}2\dim({\mathfrak{g}}/{\mathfrak{g}}^b)-\dim({\mathfrak{g}}/{\mathfrak{g}}^2)-\sum_{i=2}^{b-1}i\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1});\\ &= \frac{c}2(d-\dim({\mathfrak{g}}^b))-(d-\dim({\mathfrak{g}}^2))-\sum_{i=2}^{b-1}i\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1}); \end{align*} for |$i\ge 2$| we use the inequality |$\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1})\ge 1$|⁠, |$\dim({\mathfrak{g}}^i)\ge c-i+1$| for |$1\le i\le c$| (applied for |$i=2$| and |$i=b$|⁠) and get \begin{align*} k&\le \frac{c}2(d-c+b-1)-(d-c+1)-\frac{b(b-1)}2+1; \end{align*} write |$c=2b-e$| with |$e\in\{0,1\}$|⁠, this yields \[ k\le\frac{-3c^2+2(2d+3)c-8d+e}8. \] A polynomial function of the form |$-3x^2+2ax$| is maximal for |$x=a/3$|⁠, where it takes the value |$a^2/3$|⁠. Hence \[ k\le\frac{\frac13(2d+3)^2-8d+e}8=\frac16d^2-\frac12d+\frac{3+e}8. \] Now consider |$k'=k+D$| and again write |$c=2b-e$| with |$e\in\{0,1\}$|⁠. Then \begin{align*} k'&= \frac{c}2\dim{\mathfrak{g}}+\sum_{i=b}^{c}\left(i-\frac{c}2\right)\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1}) \\ &= \frac{cd}2+\sum_{i=b}^{c}\left(i-\frac{c}2\right)+\sum_{i=b}^{c}\left(i-\frac{c}2\right)(\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1})-1);\\ & \le \frac{cd}2+\sum_{i=b}^{c}\left(i-\frac{c}2\right)+\frac{c}2\sum_{i=b}^{c}(\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1})-1);\\ & = \frac{cd}2+\sum_{i=b}^{c}(i-c)+\frac{c}2\sum_{i=b}^{c}\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1});\\ & = \frac{cd}2+\frac{c(c+1)}{2}-\frac{b(b-1)}{2}-c(c-b+1)+\frac{c}2\sum_{i=b}^{c}(\dim({\mathfrak{g}}^i/{\mathfrak{g}}^{i+1})-1);\\ &= \frac{4cd-c^2+(2e-2)c+e}{8}+\frac{c}2\dim({\mathfrak{g}}^b). \end{align*} For all |$i\ge 2$| we have |$\dim{\mathfrak{g}}^i\le d-i$|⁠. Therefore we have (assuming |$c\ge 3$|⁠, so |$b\ge 2$|⁠) \[k'\le \frac{4cd-c^2+(2e-2)c+e}{8}+\frac{c}2(d-b)=\frac{8cd-3c^2-2c+e}{8}.\] A function of the form |$x\mapsto -3x^2+ax$| increases until it takes its maximum for |$x=a/6$|⁠; actually, using |$d\ge 4$|⁠, |$d-1\le (8d-7e-6)/6$|⁠, and since |$c\le d-1$|⁠, so the last expression is bounded above by the same when |$c$| is replaced with its maximum possible value |$d-1$|⁠. Hence \[k'\le \frac{8(d-1)d-3(d-1)^2-2(d-1)+e}{8}=\frac{5d^2-4d+e-1}{8}.\] ■ 6 The Strategy for Lower Bounds We consider a fd |$c$|-step nilpotent real Lie algebra |${\mathfrak{g}}$|⁠, with lower central series |$({\mathfrak{g}}^i)_{i\ge 1}$| and dimension |$d$|⁠. We decompose it as a direct sum of subspaces |${\mathfrak{g}}=\bigoplus_{i=1}^c\mathfrak{v}_i$|⁠, so that for all |$i$|⁠, |${\mathfrak{g}}^i=\bigoplus_{j\ge i}\mathfrak{v}_j$| (we call this a compatible decomposition). Let us choose a basis of |${\mathfrak{g}}$| as a concatenation of bases of the |$\mathfrak{v}_i$|⁠. The basis determines the subspaces |$\mathfrak{v}_i$|⁠, which is spanned by the subset of basis elements that belong to |${\mathfrak{g}}^i\smallsetminus{\mathfrak{g}}^{i+1}$|⁠. It is convenient to also directly define such a basis (without defining the |$\mathfrak{v}_i$| beforehand). A basis |$(e_1,\dots,e_d)$| in a fd nilpotent Lie algebra is compatible if it satisfies the following three conditions: for all |$i,j$|⁠, (⁠|$e_j\in{\mathfrak{g}}^i\smallsetminus{\mathfrak{g}}^{i+1}$| and |$e_k\in{\mathfrak{g}}^{i+1}$|⁠) implies |$j<k$|⁠; |$\{e_j:j\ge 1\}\cap{\mathfrak{g}}^i$| spans |${\mathfrak{g}}^i$| for all |$i$|⁠; the subspaces |${\mathfrak{g}}_i=\langle e_j:j\ge 1,e_j\in{\mathfrak{g}}^i\smallsetminus{\mathfrak{g}}^{i+1}\rangle$| span their direct sum. Thus any compatible basis determines a compatible decomposition, and any compatible decomposition yields compatible bases. In the real case, it is convenient to assume that the nonzero structure constants of |${\mathfrak{g}}$| with respect to this basis have absolute value |$\ge 1$| (this is a mild assumption as it always hold after replacing the basis by a scalar multiple, thus not changing the compatible decomposition). We define a compatible flag as a sequence of ideals \begin{equation}{\mathfrak{g}}=\mathfrak{w}_1>\mathfrak{w}_2>\dots>\mathfrak{w}_k=\{0\},\end{equation} (6.1) such that each term |${\mathfrak{g}}^i$| occurs among the |$\mathfrak{w}_j$|⁠. 6.1 Dilations Fix a nilpotent fd Lie algebra with a compatible decomposition |${\mathfrak{g}}=\bigoplus_{i=1}^c\mathfrak{v}_i$|⁠. For a nonzero scalar |$r$| define the diagonal linear automorphism |$u(r)$| of |${\mathfrak{g}}$| given by multiplication by |$r^i$| on |$\mathfrak{v}_i$|⁠. We define a new Lie algebra |${\mathfrak{g}}[r]$|⁠, with underlying |$k$|-linear space |${\mathfrak{g}}$|⁠, with the bracket |$[x,y]_r=u(r)^{-1}[u(r)x,u(r)y]$|⁠, that is, the pull-back of the original bracket by the linear automorphism |$u(r)$|⁠. Example 6.1. Consider the five-dimensional Lie algebra with basis |$(e_1,\dots,e_5)$| and nonzero brackets |$[e_1,e_3]=e_4$|⁠, |$[e_1,e_4]=[e_2,e_3]=e_5$|⁠. This is a compatible basis, with |$\mathfrak{v}_1,\mathfrak{v}_2,\mathfrak{v}_3$| having bases |$(e_1,e_2,e_3)$|⁠, |$(e_4)$| and |$(e_5)$| respectively. Then |${\mathfrak{g}}[r]$| has, in this basis, the nonzero brackets |$[e_1,e_3]_r=e_4$|⁠, |$[e_1,e_4]_r=e_5$|⁠, and |$[e_2,e_3]_r=r^{-1}e_5$|⁠. Remark 6.2. If |${\mathfrak{g}}=\bigoplus\mathfrak{v}_i$| is a Lie algebra grading (and thus a Carnot grading), then |$[\cdot,\cdot]_r=[\cdot,\cdot]$|⁠. Beware that for an arbitrary (e.g., Carnot) nilpotent Lie algebra of nilpotency length |$c\ge 3$|⁠, we can find a compatible decomposition |${\mathfrak{g}}=\bigoplus\mathfrak{v}_i$| such that |$[\mathfrak{v}_1,\mathfrak{v}_1]$| is not contained in |$\mathfrak{v}_2$|⁠. Remark 6.3. We have |${\mathfrak{g}}={\mathfrak{g}}[1]$|⁠, and the function |$(g,h,r)\mapsto [g,h]_r$| is polynomial with respect to |$r^{-1}$| (with each coefficient a skew-symmetric bilinear map |${\mathfrak{g}}\times{\mathfrak{g}}\to{\mathfrak{g}}$|⁠). The constant coefficient |$[\cdot,\cdot]_\infty$| of this polynomial can be thought as the limit of the brackets |$[\cdot,\cdot]_r$| when |$r\to\infty$| (this is indeed the case when |$K$| is a normed field), and |$({\mathfrak{g}},[\cdot,\cdot]_\infty)$| is the associated Carnot-graded Lie algebra of |${\mathfrak{g}}$|⁠, in which for |$x\in\mathfrak{v}_i$| and |$y\in\mathfrak{v}_j$|⁠, the bracket |$[x,y]_\infty$| is defined as the projection of |$[x,y]$| on |$\mathfrak{v}_{i+j}$| modulo |${\mathfrak{g}}^{i+j+1}$|⁠. This 1-parameter family of brackets has been used by Pansu and Breuillard [7, 17]. 6.2 Renormalization of the algebraic systolic growth As in Section 6.1, we fix a nilpotent fd Lie algebra with a compatible decomposition |${\mathfrak{g}}=\bigoplus_{i=1}^c\mathfrak{v}_i$|⁠, and assume in addition that the ground field is the field of real numbers. Then the dilation |$u(r)$| multiplies the Lebesgue measure (for any choice of normalization) by |$|r|^D$|⁠, where |$D$| is the homogeneous dimension of |${\mathfrak{g}}$| (see Section 2.1). We fix a norm on |${\mathfrak{g}}$| which is the sup-norm with respect to this decomposition, that is, for every |$x=\sum x_i$|⁠, |$x_i\in\mathfrak{v}_i$|⁠, we have |$\|x\|=\max\|x_i\|$|⁠. This implies that |$u(r)$| multiplies the Guivarch length by |$|r|$|⁠. This also implies that for |$x\in{\mathfrak{g}}$|⁠, the conditions |$\|x\|\ge 1$| and |${\lfloor}loor x\rfloor\ge 1$| are equivalent. Let |$\Lambda_r$| be a discrete cocompact subring of Guivarch systole |$\ge r$|⁠. For instance, assuming that |${\mathfrak{g}}$| admits a rational structure, we can choose, by compactness, |$\Lambda_r$| to have covolume exactly |$\sigma(r)$| (i.e., has minimal covolume among those discrete cocompact subrings of Guivarch systole |$\ge r$|⁠. Then |$\Xi_r=u(r)^{-1}\Lambda_r$| has Guivarch systole |$\ge 1$|⁠; this is a discrete cocompact subring in |${\mathfrak{g}}[r]$|⁠. Its covolume satisfies |${\mathrm{cov}}(\Xi_r)=r^{-D}{\mathrm{cov}}(\Lambda_r)$|⁠. The idea is that rather than studying |$\Lambda_r$| (whose systole tends to infinity), we study |$\Xi_r$| (which has bounded systole |$\ge 1$|⁠), the only caveat being that |$\Xi_r$| is a Lie subring of |${\mathfrak{g}}[r]$| (i.e., for some Lie bracket which varies with |$r$|⁠). 6.3 First application: small systolic growth Assume here that |$\underline{\lim}r^{-D}\sigma(r)<\infty$| and let us prove that |${\mathfrak{g}}$| is Carnot. Suppose that |${\mathrm{cov}}(\Lambda_r)\le Cr^D$| (for |$r\in I$|⁠, where |$I$| is an unbounded set of positive real numbers). Then |${\mathrm{cov}}(\Xi_r)\le C$|⁠. By compactness of the set of lattices with systole |$\ge 1$| and covolume |$\le C$|⁠, we can find an unbounded subset |$J\subset I$| such that |$\lim_{J\ni r\to\infty}\Xi_r=\Xi_\infty$| (in the Chabauty topology) for some lattice |$\Xi_\infty$|⁠. Clearly |$\Xi_\infty$| is a subring of |$({\mathfrak{g}},[\cdot,\cdot]_\infty)$|⁠. Then |$\Xi_r$| is isomorphic as a Lie ring to |$\Xi_\infty$| for large enough |$r\in J$|⁠. Indeed, choose a basis |$(u_1,\dots,u_d)$| of |$\Xi_\infty$| as a |$\mathbf{Z}$|-module. Then we can find |$u_i^r\in\Xi_r$| with |$u_i^r\to u_i$|⁠. Then |$[u_i,u_j]=\sum_kn_{ij}^ku_k$| for suitable integers |$n_{ij}^k$| (here exponents are additional indices, and not powers). Then |$[u_i^r,u_j^r]-\sum_kn_{ij}^ku_k^r$| belongs to |$\Xi_r$| and converges to zero, hence is zero for |$r$| large enough. Moreover, since the covolume is a continuous function, eventually |$(u_i^r)$| is a |$\mathbf{Z}$|-basis of |$\Xi_r$|⁠. This shows that this is indeed an isomorphism. Since |${\mathfrak{g}}\simeq\Xi_r\otimes_{\mathbf{Z}}\mathbf{R}$| and |$({\mathfrak{g}},[\cdot,\cdot]_\infty)\simeq\Xi_\infty\otimes_{\mathbf{Z}}\mathbf{R}$| as real Lie algebras, this shows that |$({\mathfrak{g}},[\cdot,\cdot]_\infty)\simeq{\mathfrak{g}}$|⁠. Thus |${\mathfrak{g}}$| is Carnot. 6.4 Covolume inequalities (In this Section 6.4, the Lie algebra bracket plays no role.) Assume now that the ground field is the field of real numbers. We fix a compatible flag as in (6.1), denoted |$F$| for short. We fix Lebesgue measures on all |$\mathfrak{w}_j/\mathfrak{w}_{j-1}$|⁠, and thus on |${\mathfrak{g}}$|⁠. Fix a compatible basis, compatible with this flag. This basis defines (in a compatible way) normalizations of the Lebesgue measures on each quotient |$\mathfrak{w}_j/\mathfrak{w}_k$|⁠, where the cube |$[0,1\mathclose[^q$| (for the given basis) has volume 1, and sup norms with respect to this basis. The basis also yields a compatible decomposition; we thus have a notion of Guivarch length (Section 2.1). Observe that |$u(r)$| multiplies the Guivarch length by |$|r|$|⁠. We say that an additive lattice |$\Lambda$| of |$({\mathfrak{g}},+)$| is |$F$|-compatible if |$\Lambda\cap\mathfrak{w}_j$| is a lattice in |$\mathfrak{w}_j$| for every |$j$|⁠. This implies that for all |$j$| the projection |$\Lambda_{[j]}$| of |$\Lambda\cap\mathfrak{w}_j$| on |$\mathfrak{f}_j=\mathfrak{w}_j/\mathfrak{w}_{j-1}$| is a lattice in |$\mathfrak{f}_j$| as well. Let |$a_j(\Lambda)$| be the covolume of |$\Lambda_{[j]}$| in |$\mathfrak{f}_j$|⁠. Then the covolume of |$\Lambda\cap\mathfrak{w}_j$| is equal to |$\prod_{k\ge j}a_k(\Lambda)$|⁠. For any |$v\in{\mathfrak{g}}$|⁠, we have |$\|v\|\ge 1$| if and only if |${\lfloor}loor v\rfloor\ge 1$|⁠. So for an additive lattice of systole |$\ge 1$| (for either the norm or the Guivarch length), the covolume is |$\ge 1$|⁠. Thus, |$\prod_{k\ge j}a_k(\Lambda)\ge 1$| for all |$j$|⁠. This means that the following holds: Lemma 6.4. For any |$j$|⁠, any |$F$|-compatible additive lattice of |${\mathfrak{g}}$| of Guivarch systole |$\ge 1$| satisfies \[ \prod_{k\ge j}a_k(\Lambda)\ge 1,\quad\text{or equivalently,}\quad \sum_{k\ge j}\log_r(a_k(\Lambda))\ge 0,\quad\forall r>1. \] In the sequel, these inequalities will be combined with other inequalities making use of further assumptions (namely that |$\Lambda$| is a Lie subring). The requirements about the norm will be fulfilled when we consider a Lie algebra with basis |$(e_1,\dots,e_d)$|⁠, and a flag containing all elements of the lower central series such that, denoting |${\mathfrak{g}}_{\ge i}$|⁠, each element of the flag is one of the |${\mathfrak{g}}_{\ge i}$|⁠; on each |${\mathfrak{g}}_{\ge i}/{\mathfrak{g}}_{\ge j}$| the norm being the |$\ell^\infty$| norm and the Lebesgue measure being normalized so that the cube |$[0,1\mathclose[^k$| has measure 1. 7 Precise Estimates We give explicit estimates of the systolic growth for various illustrating examples. While we obtain upper bounds by easy explicit construction, we use the method of Section 6 to obtain lower bounds. 7.1 Five-dimensional non-Carnot Lie algebras Following a convenient custom, when we describe a Lie algebra by saying that the nonzero brackets are |$[e_i,e_j]=f_{ij}$|⁠, we mean that |$[e_j,e_i]=-f_{ij}$| and that all other brackets between basis elements are zero. That an algebra defined in such a way is indeed Lie is equivalent to say that \[J(e_i,e_j,e_k)=[e_i,[e_j,e_k]]+[e_j,[e_k,e_i]]+[e_k,[e_i,e_j]]=0,\quad \forall i<j<k.\] There are exactly two non-isomorphic non-Carnot nilpotent five-dimensional real Lie algebras. Using the notation in [10], these are defined, in the basis |$(e_1,\dots,e_5)$|⁠, by the nonzero brackets: \begin{align*} L_{5,5}:\;& \qquad [e_1,e_2]=e_4, && [e_1,e_4]=e_5, && [e_2,e_3]=e_5;\\ L_{5,6}:\;& [e_1,e_2]=e_3, \;[e_1,e_3]=e_4,&& [e_1,e_4]=e_5, && [e_2,e_3]=e_5. \end{align*} The lower central series in |$L_{5,5}$| is |$123/4/5$| (this concise notation means |${\mathfrak{g}}^2={\mathfrak{g}}_{\ge 4}$| and |${\mathfrak{g}}^3={\mathfrak{g}}_{\ge 5}$|⁠) and in |$L_{5,6}$| it is |$12/3/4/5$|⁠. Thus we can write the Lie algebra law of |${\mathfrak{g}}[r]$| in each case: \begin{align*} L_{5,5}:\;& \qquad [e_1,e_2]_r=e_4, && [e_1,e_4]_r=e_5, && [e_2,e_3]_r=r^{-1}e_5;\\ L_{5,6}:\;& [e_1,e_2]_r=e_3,\; [e_1,e_3]_r=e_4,&& [e_1,e_4]_r=e_5, && [e_2,e_3]_r=r^{-1}e_5. \end{align*} Lemma 7.1. In both cases, the complete flag |$({\mathfrak{g}}_{\ge i})_{1\le i\le 5}$| is made up of solid ideals. Proof. All are part of the lower central series, except |${\mathfrak{g}}_{\ge 2}$| in both cases and |${\mathfrak{g}}_{\ge 3}$| for |$L_{5,5}$|⁠. The upper central series is |$12/34/5$| for |$L_{5,5}$|⁠, so |${\mathfrak{g}}_{\ge 3}$| is also solid in this case. Finally, |${\mathfrak{g}}_{\ge 2}$| is the centralizer of |${\mathfrak{g}}_{\ge 4}$| in both cases, so is solid. ■ Let now, in either case, |$\Lambda_r$| be a discrete cocompact subring in |${\mathfrak{g}}$|⁠, with Guivarch systole |$\ge r$| and covolume |$\sigma(r)$|⁠, and define |$\Xi_r=u(r)^{-1}\Lambda_r$|⁠, which is a discrete cocompact subring of |${\mathfrak{g}}[r]$| with systole |$\ge 1$|⁠. As in Section 6.4, let |$a_i(r)$| be the systole of the projection of |$\Xi_r\cap{\mathfrak{g}}^i$| in |${\mathfrak{g}}^i/{\mathfrak{g}}^{i+1}$|⁠, and write |$A_i=\log_r(a_i(r))$|⁠. Lemma 7.2. In both cases, we have, for all |$r>0$|⁠, the inequalities |$A_1+A_4\ge 0$|⁠, |$A_2+A_3\ge 1$|⁠, |$A_5\ge 0$|⁠. In particular, |$\sum_{i=1}^5A_i\ge 1$|⁠. Proof. It is convenient to denote by |$o(i)$|⁠, resp. |$O(i)$|⁠, an unspecified element of |${\mathfrak{g}}_{\ge i+1}$|⁠, resp. |${\mathfrak{g}}_{\ge i}$|⁠. By definition, |$\Xi_r$| contains elements |$v_1=a_1e_1+o(1)$|⁠, |$v_2=a_2e_2+o(2)$|⁠, |$v_3=a_3e_3+o(3)$|⁠, |$v_4=a_4e_4+o(4)$|⁠, |$v_5=a_5e_5$|⁠. Then |$A_5\ge 0$| means |$a_5=\|v_5\|\ge 1$|⁠, which is a trivial consequence of having systole |$\ge 1$|⁠. Then |$[o(1),O(4)]=[O(1),o(4)]=[o(2),O(3)]=[O(2),o(3)]=0$| in both cases. Since for both |$L_{5,5}[r]$| and |$L_{5,6}[r]$| we have |$[e_1,e_4]_r=e_5$| and |$[e_2,e_3]_r=r^{-1}e_5$|⁠, it follows that |$[v_1,v_4]_r=a_1a_4e_5$| and |$[v_2,v_3]_r=r^{-1}a_2a_3e_5$|⁠. Therefore |$a_1a_4\ge 1$| and |$r^{-1}a_2a_3\ge 1$|⁠, which means that |$A_1+A_4\ge 0$| and |$A_2+A_3\ge 1$|⁠. The last inequality follows \[ A_1+A_2+A_3+A_4+A_5= (A_1+A_4)+(A_2+A_3)+A_5\ge 0+1+0=1. \] ■ Lemma 7.2 shows that the covolume |${\mathrm{cov}}(\Lambda_r)=\prod_{i=1}a_i$| is |$\ge r$|⁠, which in turn means that |$\sigma(r)={\mathrm{cov}}(\Lambda_r)=r^D{\mathrm{cov}}(\Xi_r)\ge r^{D+1}$|⁠. To get a reverse inequality, let us define |$\Xi'_r$| as the additive lattice with basis |$(e_1,re_2,e_3,e_4,e_5)$|⁠. This is indeed a subring of |$({\mathfrak{g}},[\cdot,\cdot]_r)$|⁠. Its covolume is |$r$| and its systole is |$\ge 1$|⁠. We thus define |$\Lambda'_r=u(r)\Xi'_r$| (an explicit basis depends on |$u(r)$|⁠, which is not the same for the two Lie algebras in consideration): it has Guivarch systole |$r$| and covolume |$r^{D+1}$|⁠. Accordingly, for both Lie algebras, for this choice of norm, compatible decomposition and covolume, |$\sigma(r)=r^{D+1}$|⁠. Actually, the previous construction of lattices |$(\Xi'_r)$| satisfies some additional features: first, when |$n$| is a positive integer, |$\Lambda'_n$| is contained in the subring |${\mathfrak{g}}[\mathbf{Z}]$|⁠. This implies that the systolic growth of the corresponding lattices is |$\preceq r^{D+1}$|⁠. Since real nilpotent Lie algebra up to dimension 5 have a unique rational structure up to automorphism (see [10]), this implies that the systolic growth of the lattices is also |$\simeq r^{D+1}$|⁠, proving 1.4. Second, the |$G$|-uniform Guivarch systole of the lattices |$\Xi'_r$| is also |$\ge 1$|⁠: this is because every nonzero element has the form |$ne_i+o(i)$| for some integer |$n$| and hence all its |$G$|-conjugates still has the form |$ne_i+o(i)$| and thus has Guivarch norm |$\ge 1$|⁠. Therefore, this proves that the |$G$|-uniform systolic growth of |$G$| and its lattices is |$\simeq r^{D+1}$| as well. 7.2 One example with non-integral polynomial degree Its law is given by the symbolic notation \begin{gather*} 12|3,\;13|4,\;14|5,\;15|6,\; 16|7,\\ 23|5,\;24|6,\; 34|7. \end{gather*} This means that the nonzero brackets are given by |$[e_1,e_2]=e_3$|⁠, |$[e_1,e_3]=e_4$|⁠, etc. (It appears as |${\mathfrak{g}}_{7,1,1(0)}$| in Magnin’s classification [16]). It is filiform, that is, its nilpotency length is as large as possible for the given dimension; its lower central series is given, in symbols, as |$12/3/4/5/6/7$|⁠. All the |${\mathfrak{g}}_{\ge i}$| are solid (see Example 2.7). Once more, let |$\Lambda_r$| be a discrete cocompact subring of covolume |$\le r$| and Guivarch systole |$\sigma(r)$|⁠, |$\Xi_r=u(r)^{-1}\Lambda_r$|⁠. We define |$a_i,A_i$| as previously. Then, by Lemma 6.4, \[\sum_{i=j}^7A_i\ge 0,\quad 1\le j\le 7\] Also the law in |$\Lambda_r$| is given by \begin{gather*} 12|3,\;13|4,\;14|5,\;15|6,\; 16|7,\\ 23|r^{-1}5,\;24|r^{-1}6,\; 34|r^{-1}7. \end{gather*} Lemma 7.3. |$2A_1+A_5+A_6\ge 0$| and |$2(A_2+A_3+A_4)\ge 3$|⁠. Proof. There exist elements |$v_i=a_ie_i+o(i)$| in |$\Lambda_r$|⁠. Then |$[v_1,v_5]=a_1a_5e_6+o(6)$| and |$[v_1,v_6]=a_1a_6e_7$|⁠. These two elements generate a lattice of covolume |$a_1^2a_5a_6$| in |${\mathfrak{g}}_{\ge 6}$|⁠. Since it has systole |$\ge 1$|⁠, we deduce |$a_1^2a_5a_6\ge 1$|⁠, that is, |$2A_1+A_5+A_6\ge 0$|⁠; Next |$[v_2,v_3]=r^{-1}a_2a_3e_5+o(5)$|⁠, |$[v_2,v_4]=r^{-1}a_2a_4e_6+o(6)$|⁠, |$[v_3,v_4]=r^{-1}a_3a_4e_7$|⁠. These three elements generate a lattice of covolume |$r^{-3}a_2^2a_3^3a_4^2$| in |${\mathfrak{g}}_{\ge 5}$|⁠. Since it has systole |$\ge 1$|⁠, we deduce |$r^{-3}a_2^2a_3^3a_4^2\ge 1$|⁠, and hence |$2(A_2+A_3+A_4)\ge 3$|⁠. ■ Corollary 7.4. |$\sum A_i\ge 3/2$|⁠. Proof. Indeed, |$A_5+A_6+A_7\ge 0$| by Lemma 6.4 and hence \[2\sum A_i=2(A_2+A_3+A_4)+(2A_1+A_5+A_6)+(A_5+A_6+A_7)+A_7\ge 3.\] ■ We deduce |$\sigma(r)\preceq n^{D+3/2}$| (here |$D=2+\sum_2^7k=29$|⁠). In the other direction, for |$r\ge 1$| we define |$\Xi'_r$| to be the lattice with basis |$(e_1,\sqrt{r}e_2,\sqrt{r}e_3,\sqrt{r}e_4,e_5,e_6,e_7)$|⁠. This is a discrete cocompact subring in |${\mathfrak{g}}[r]$|⁠, with Guivarch systole |$1$| and covolume |$r^{3/2}$|⁠. So |$\Lambda'_r=u(r)M_r$| is a discrete cocompact subring of Guivarch systole |$r$| and covolume |$r^{D+3/2}$|⁠, so that |$\sigma(r)\preceq r^{D+3/2}$|⁠. To conclude, |$\sigma(r)\simeq r^{D+3/2}$| (with |$D=29$|⁠). Moreover, for |$r$| a square integer, this lattice |$\Lambda'_r$| has an integral basis, so this also provides an upper bound for the systolic growth of the lattices relative to the given rational structure. 7.3 Truncated Witt Lie algebras Here we prove Theorem 1.6. The method is similar to the approach in the previous examples, except the final computation, which is more complicated. So, let us begin with this computation and briefly make the connection afterwards. Lemma 7.5. Consider the system of inequalities, with real unknowns |$A_1,\dots,A_n$|⁠: \[\begin{cases} \sum_{i=j}^nA_i\ge 0\qquad\forall j \\ A_1+A_i\ge A_{i+1}\qquad\forall i=2,\dots,n-1 \\ A_i+A_j\ge A_{i+j}+1,\qquad\forall 2\le i<j,\;i+j\le n \end{cases}\] Then under this constraints, we have |$\sum_{i=1}^n A_i\ge\lceil (n-4)/2\rceil$|⁠, and this is attained when we set |$A_i=1$| for |$2\le i<n/2$|⁠, and |$A_i=0$| for other |$i$| (⁠|$i=1$| and |$n/2\le i\le n$|⁠). Proof. The given solution realizes the inequalities and the claimed minimal value of |$\sum A_i$|⁠. Let us show the lower bound on |$\sum A_i$|⁠. We begin with |$n=2m-1$| odd (⁠|$m\ge 2$|⁠), so we have to show |$\sum A_i\ge m-2$|⁠. Then \begin{align*} \sum_{i=1}^{2m-1} A_i &= A_1+A_{2m-2}+A_{2m-1}+\sum_{i=2}^{m-1}(A_i+A_{2m-1-i})\\ & \ge 2A_{2m-1}+\sum_{i=2}^{m-1}(A_{2m-1}+1)=m-2+mA_{2m-1}\ge m-2. \end{align*} The case when |$n=2m$| is even (where we have to prove |$\sum A_i\ge m-2$|⁠) is a bit more complicated as we gather terms by triples, which leads to discuss on the value of |$m$| modulo 3. Let |$i\ge 1$| be such that |$m-3i\ge 2$| (so |$m+3i-1\le 2m-3$|⁠). We consider the following sum of |$6i$| terms \begin{align} \sum_{j=m-3i}^{m+3i-1}A_i &= \sum_{k=1}^i (A_{m-3k}+A_{m-3k+1}+A_{m-3k+2}+A_{m+3k-3}+A_{m+3k-2}+A_{m+3k-1})\notag\\ &= \sum_{k=1}^i (A_{m-3k}+A_{m+3k-2})+(A_{m-3k+1}+A_{m+3k-1})+(A_{m-3k+2}+A_{m+3k-3})\notag\\ &\ge \sum_{k=1}^i (1+A_{2m-2})+(1+A_{2m})+(1+A_{2m-1})\notag\\ &= 3i+i(A_{2m-2}+A_{2m-1}+A_{2m}). \end{align} (7.1) If instead, we choose |$i$| such that |$m-3i=1$|⁠, computing the previous inequality works in the same way, except that once we have to use an inequality of the form |$A_1+A_j\ge A_{j+1}$| (instead of |$A_1+A_j\ge A_{j+1}+1$|⁠), so there is one less |$+1$| term and we get \begin{equation} \sum_{j=m-3i}^{m+3i-1}A_i\ge (3i-1)+i(A_{2m-2}+A_{2m-1}+A_{2m}). \end{equation} (7.2) If |$m=3k+1$|⁠, we choose |$i=k$|⁠, so that |$(m-3i,m+3i-1)=(1,2m-2)$|⁠, |$3i-1=m-2$|⁠; we then have, using (7.2) \[\sum_{i=1}^{2m} A_i \ge (A_{2m-1}+A_{2m})+ m-2+k(A_{2m-2}+A_{2m-1}+A_{2m}) \ge m-2.\] If |$m=3k$|⁠, we choose |$i=k-1$|⁠, so that |$(m-3i,m+3i-1)=(3,2m-4)$|⁠, |$3i=m-3$|⁠; using (7.1) we get. \begin{align*} \sum_{i=1}^{2m} A_i \ge & (A_1+A_{2m-1})+(A_2+A_{2m-3})+A_{2m-2}+A_{2m}\\ & + m-3+(k-1)(A_{2m-2}+A_{2m-1}+A_{2m})\\ \ge & m-2+A_{2m}+k(A_{2m-2}+A_{2m-1}+A_{2m})\ge m-2. \end{align*} Finally if |$m=3k+2$|⁠, we choose |$i=k$|⁠, so |$(m-3i,m+3i-1)=(2,2m-3)$|⁠, |$3i=m-2$|⁠. Then, using that \begin{align*} A_1+A_{2m-2}+A_{2m-1} &= \frac12((A_1+A_{2m-2})+(A_1+A_{2m-1})+(A_{2m-2}+A_{2m-1})\\ &\ge \frac12(A_{2m-1}+A_{2m}+(A_{2m-2}+A_{2m-1})\\ &=\frac12(A_{2m-2}+2A_{2m-1}+A_{2m}), \end{align*} we get, incorporating the previous inequality (7.1), \begin{align*} \sum_{i=1}^{2m} A_i & \ge A_1+A_{2m-2}+A_{2m-1}+A_{2m}+ m-2+i(A_{2m-2}+A_{2m-1}+A_{2m})\\ & \ge \frac12A_{2m}+\frac12(A_{2m-1}+A_{2m})+(i+1/2)(A_{2m-2}+A_{2m-1}+A_{2m})+m-2\\ & \ge m-2. \end{align*} ■ To prove Theorem 1.6, we can assume |$n\ge 4$|⁠, so that |${\mathfrak{g}}_{\ge i}$| are solid, as checked in Example 2.7. Then for convenience we rescale the basis by considering |$f_i=n^{-1}e_i$|⁠. Then |$[\,f_i,f_j]=\frac{(i-j)}{n}f_{i+j}$| for |$i+j\le n$|⁠, so the coefficients are all |$\le 1$|⁠, and we consider the |$\ell^\infty$| norm with respect to this basis. It follows that the bracket in |${\mathfrak{g}}[r]$| is given by \[[\,f_1,f_i]=\frac{1-i}{n}f_{i+1}\;(i\ge 2);\quad [\,f_i,f_j]=\frac{r^{-1}(i-j)}{n}f_{i+j}\; (i,j\ge 2).\] As in the previous cases, we consider a discrete cocompact subring |$\Lambda_r$| of Guivarch systole |$\ge r$| and |$\Xi_r=u(r)^{-1}\Lambda_r$|⁠, which has systole |$\ge 1$| and is a discrete cocompact subring in |${\mathfrak{g}}[r]$|⁠. If |$a_i$| is the systole of the projection on |${\mathfrak{g}}_{\ge i}/{\mathfrak{g}}_{\ge i+1}$|⁠, we consider an element |$a_if_i+o(i)$| for all |$i$|⁠. Computing the brackets and defining |$A_i=\log_r(a_i)$|⁠, we deduce |$A_1+A_i\ge A_{i+1}$| (⁠|$2\le i\le n-1$|⁠) and |$A_i+A_j\ge A_{i+j}+1$| (⁠|$2\le i<j\le n-i$|⁠) (here we use that nonzero structure coefficients are |$\le 1$| in absolute value). This gives rise to the system of inequalities solved in Lemma 7.5, and hence |$\sum A_i\ge{\lfloor}loor (n-4)/2\rfloor$|⁠. Accordingly, the covolume of |$\Xi_r$| is |$\ge r^{{\lfloor}loor (n-4)/2\rfloor}$|⁠, and hence that of |$\Lambda_r$| is |$\ge r^{D+{\lfloor}loor (n-4)/2\rfloor}$|⁠. (The case |$n=5$| covers |$L_{5,6}$| from Section 7.1). 7.4 Further examples The next example, unlike the previous ones, do not admit a complete flag of solid ideals. Therefore, lower bounds on the systolic growth now rely on some geometric lemmas about lattices, such as Lemma 7.7 below. Consider the Lie algebra |${\mathfrak{g}}={\mathfrak{g}}(2n+4)$| (here |$2n+4$| is the dimension) with basis \[(U,V,W,Z,X_1,Y_1,\dots,X_n,Y_n)\] and nonzero brackets \[[U,V]=W,\quad [U,W]=Z,\quad [X_i,Y_i]=Z, \; \forall 1\le i\le n.\] Its nilpotency length is 3, the derived subalgebra |${\mathfrak{g}}^2$| is the plane generated by |$(W,Z)$|⁠, |${\mathfrak{g}}^3$| is the line generated by |$Z$|⁠. We have |$\dim({\mathfrak{g}}/{\mathfrak{g}}^2)=2n+2$|⁠, and the homogeneous dimension is |$D=2n+7$|⁠. Proposition 5.1 predicts |$\sigma(r)\preceq r^{D+n+1}$|⁠. This is not sharp but almost: Theorem 7.6. For every fixed |$n$|⁠, the systolic growth of |${\mathfrak{g}}(2n+4)$|⁠, as a function of |$r$|⁠, is |$\simeq r^{3n+7}=r^{D+n}$|⁠. The proof relies on the following general lemma. Lemma 7.7. Let |$V$| be a |$d$|-dimensional real vector space (⁠|$d$| even) with a fixed Lebesgue measure. Let |$\phi$| be a symplectic form of determinant 1. Let |$\Gamma$| be a lattice in |$V$| such that for all |$x,y\in\Gamma$| we have |$\phi(x,y)\in\mathbf{Z}$|⁠. Then the covolume of |$\Gamma$| is |$\ge 1$|⁠. If for some |$s>0$|⁠, we have |$\phi(x,y)\in s\mathbf{Z}$| for all |$x,y\in\Gamma$|⁠, then the covolume of |$\Gamma$| satisfies |${\mathrm{cov}}(\Gamma)\ge s^{d/2}$|⁠. Proof. We argue by induction on |$d/2$|⁠. The case |$d=0$| is trivial. Fix a primitive element |$e_1$| in |$\Gamma$|⁠. Let |$H$| be its orthogonal for |$\phi$| and |$H'=H/\mathbf{R} e_1$|⁠. Then |$\phi(e_1,\Gamma)$| is a nonzero subgroup |$m\mathbf{Z}$| of |$\mathbf{Z}$|⁠, with |$m\ge 1$|⁠. Let |$e_d$| be an element in |$\Gamma$| with |$\phi(e_1,e_d)=m$|⁠. Every element in |$x\in\Gamma$| can be written in a unique way as |$\lambda_x e_d+h_x$| with |$(\lambda_x,hx)\in\mathbf{R}\times H$|⁠. Then |$\phi(e_1,x)=\lambda_x m$| belongs to |$m\mathbf{Z}$|⁠; this shows that |$\lambda_x\in\mathbf{Z}$|⁠, and hence |$h_x\in\Gamma$| as well. This shows that |$\Gamma=(\Gamma\cap H)\oplus \mathbf{Z} e_d$|⁠. Since |$e_1$| is primitive, we can write |$\Gamma\cap H=\mathbf{Z} e_1\oplus \Gamma'$|⁠, where |$\Gamma'$| is a lattice in a hyperplane |$V'$| of |$H$|⁠. We fix Lebesgue measures on |$V'$| and on |$(\mathbf{R} e_1\oplus \mathbf{R} e_d)$| so that the both restrictions of |$\phi$| to these subspaces has determinant 1. So the product measure matches with the original Lebesgue measure on |$V$|⁠. By induction, the covolume |$c'$| of |$\Gamma'$| in |$V'$| is |$\ge 1$|⁠, and the covolume of |$\mathbf{Z} e_1\oplus \mathbf{Z} e_d$| in the plane it spans is |$m$|⁠. So the covolume of |$\Gamma$| is |$c'm\ge 1$|⁠. For the second result, we apply the hypothesis to the lattice |$s^{-1/2}\Gamma$|⁠, which has covolume |$\ge 1$|⁠; its covolume is also |$s^{-d/2}{\mathrm{cov}}(\Gamma)$|⁠. This proves that |${\mathrm{cov}}(\Gamma)\ge s^{d/2}$|⁠. ■ Proof of Theorem 7.6. First define |$\Lambda_r$| as the lattice with basis |$rU$|⁠, |$rV$|⁠, |$rX_i$|⁠, |$r^2Y_i$| (⁠|$1\le i\le n$|⁠), |$r^2W$|⁠, |$r^3Z$|⁠. Its covolume is |$r^{7+3n}$|⁠. Its systole is |$r$|⁠. It is a subring, by a straightforward verification. So the systolic growth is |$\preceq r^{7+3n}$|⁠. We have just described the lower central series. Also, the center is reduced to the line generated by |$Z$| and the second term in the ascending central series is the codimension 2 subspace |$\mathfrak{j}$| generated by all basis vectors except |$U,V$|⁠. The centralizer |$\mathfrak{c}$| of |$\mathfrak{j}$| is three-dimensional with basis |$(V,W,Z)$|⁠, and |$\mathfrak{h}=\mathfrak{c}+\mathfrak{j}$| is the hyperplane generated by all basis vectors except |$U$|⁠. So we have the inclusions of solid ideals \[\{0\}\stackrel{Z}\subset{\mathfrak{g}}^3\stackrel{W}\subset{\mathfrak{g}}^2\stackrel{V}\subset\mathfrak{c}\stackrel{X_1,\dots,Y_n}\subset\mathfrak{h}\stackrel{U}\subset{\mathfrak{g}}\] The dilation |$u(r)$| is given by multiplication by |$r$| on the subspace with basis |$(U,V,X_1,\dots,Y_n)$|⁠, by |$r^2$| on |$W$| and by |$r^3$| on |$Z$|⁠. The nonzero brackets in |${\mathfrak{g}}[r]$| are thus given by \[[U,V]_r=W,\quad [U,W]_r=Z,\quad [X_i,Y_i]_r=r^{-1}Z, \; \forall 1\le i\le n.\] Let a discrete cocompact subring |$\Lambda_r$| have systole |$\ge r$|⁠. Define |$\Xi_r=u(r)^{-1}\Lambda_r$|⁠. So |$\Xi_r$| intersects each of these ideals in a lattice, and the projection modulo the previous ideal, written in the canonical basis, gives generators |$\alpha U$|⁠, |$\beta V$|⁠, |$\delta W$|⁠, |$\eta Z$| (we choose the constant to be positive) and a lattice |$\Gamma$| in the space with basis |$(X_1,\dots,Y_n)$|⁠, elements. So |$\Lambda_r$| contains an element |$u$| of the form |$\alpha U+x$| (with |$x$| a combination of the other generators), and an element |$v$| of the form |$\beta V+tW+t'Z$|⁠, and an element |$w$| of the form |$\delta W+t''Z$|⁠. Then |$[u,v]=\alpha\beta W+\alpha t Z$|⁠, and |$[u,w]=\alpha\delta Z$|⁠. Since the intersection of |$\Lambda_r$| with the plane with basis |$(W,Z)$| has covolume |$\delta\eta$| and the covolume of its subgroup generated by |$[u,v]$| and |$[u,w]$| is |$\alpha^2\beta\delta$|⁠, we get |$\alpha^2\beta\delta\ge\delta\eta$|⁠, or equivalently |$\alpha^2\beta\ge\eta$|⁠. Lemma 6.4 implies that all of |$\beta\delta\eta$|⁠, |$\delta\eta$|⁠, |$\eta$| are |$\ge 1$|⁠. It does not say that |$\alpha\beta\delta\eta\ge 1$| because we have one intermediate term in the filtration, but we can deduce it: \[(\alpha\beta\delta\eta)^2=(\alpha\beta\alpha)\beta\delta^2\eta^2\ge \eta\beta\delta^2\eta^2=(\beta\delta\eta)(\delta\eta)\eta\ge 1.\] On the other hand, we see that the standard symplectic form |$\phi$| on the subspace generated by |$X_1,\dots,Y_n$| (for which |$[X_i,Y_i]=\phi(X_i,Y_i)Z$|⁠) maps |$\Gamma\times\Gamma$| into |$r\eta\mathbf{Z}$|⁠. It follows from Lemma 7.7 that |${\mathrm{cov}}(\Gamma)\ge (r\eta)^n\ge r^n$|⁠. Therefore, the covolume of |$\Xi_r$| satisfies \[{\mathrm{cov}}(\Xi_r)={\mathrm{cov}}(\Gamma)\alpha\beta\delta\eta\ge r^n.\] Thus the systolic growth satisfies |$\sigma(r)\ge r^{D+n}$|⁠. ■ Remark 7.8. The above proof can be extended to a larger class of Lie algebras, namely for |$n\ge 0$| and |$k\ge 3$|⁠, define |${\mathfrak{g}}(k+2n,k)$| with basis |$U_1,\dots,U_k$|⁠, |$ X_1,Y_1,\dots,X_n,Y_n$|⁠, and nonzero brackets |$[U_1,U_i]=U_{i+1}$| (⁠|$2\le i\le k-1$|⁠), |$[X_i,Y_i]=U_k$|⁠. Then its systolic growth is |$\simeq r\mapsto r^{D+n(k-3)}$|⁠. (For |$k=5$|⁠, the nilpotency length is 4; this yields |$\sigma(r)\simeq r^{D+2n}$| while Proposition 5.1 predicts |$\sigma(r)\preceq r^{D+2n+2})$|⁠. If we write the dimension as |$d=2n+k$| and write |$h=n(k-3)$|⁠, we see that |$h$| is maximal when |$n=(d-3)/4$|⁠, that is, for |$n$| integer, |$n=(d-3+e)/4$| with |$e$| an integer with |$|e|\le 2$|⁠. Then for such |$n$| the computation provides |$h=\frac18((d-3)^2-e^2)$|⁠, with |$e^2\in\{0,1,4\}$| (⁠|$e^2=0$| for |$d\stackrel{4}\equiv 3$|⁠, |$e^2=1$| for |$d$| even, |$e^2=4$| for |$d\stackrel{4}\equiv 1$|⁠). Also, if we choose |$\lambda=(1+\sqrt{5})/2\sim 1.618\dots$|⁠, then we see that for |${\mathfrak{g}}(n,{\lfloor}loor \lambda n\rfloor)$|⁠, we have |$h=(\alpha+O(1/n))\frac{d^2}6$|⁠, with |$\alpha=\frac{6\lambda}{(2+\lambda)^2}\sim 0.742\dots$|⁠. This shows that |$h$| can behave as fast as the square of the dimension (we choose to write it as a factor of |$\frac{d^2}6$|⁠, since |$h\le \frac{d^2}6$| by Propositions 5.1 and 5.2). Funding This material is based upon work supported by the {NSF under [Grant No. DMS-1440140]} while the author was in residence at the MSRI (Berkeley) during the Fall 2016 semester. Partially supported by ANR Project ANR-14-CE25-0004 GAMME. Acknowledgements The author thanks Yves Benoist for useful hints, and Pierre de la Harpe for corrections on a preliminary version of this article. The author also thanks the referee for various corrections and useful references. Appendix. Lattices and Discrete Cocompact Subrings: Back and Forth We prove here Lemma 4.1. This lemma was mentioned to the author by Yves Benoist. While it essentially follows from Malcev’s ideas, I could not find a reference with this statement precisely written. The closest I am aware of are in the books [18, Chap. 6] and [8, Section 5.4]: [18, Chap. 6, Theorem 5] essentially says that every lattice |$\Gamma$| is trapped between two lattices whose logarithms are additive subgroups, so that the index between the two is bounded only in terms of the dimension of the ambient Lie group. In [8, Section 5.4], the main result in the direction of Lemma 4.1 is Proposition 5.4.8 of that book, which states that every lattice |$\Gamma$| has a finite index subgroup |$\Gamma'$| such that |$\log\Gamma'$| is a subring (and furthermore, |$\log\Gamma$| is a finite union of additive cosets of |$\log\Gamma'$|⁠). Unlike in [18], no uniform bound is given. These results are not enough to prove Lemma 4.1 for two reasons: they do not yield anything in the direction (1) of Lemma 4.1; in the direction (2), they provide partial statements. In [18] it only yields subgroups whose logarithm is an additive lattice (but maybe not a Lie subring), and in [8] the uniformity not given. Let us start with a general elementary fact on the covolume. We now proceed to state a more precise and robust version of Lemma 4.1 and then prove it. Given a nilpotent Lie |$\mathbf{Q}$|-algebra, we can endow it with a group law by setting |$xy=\log(\exp(x)\exp(y))$|⁠, given by the Baker–Campbell–Hausdorff formula; we still call it multiplication and denote it by |$\cdot$| (or no sign). It is thus endowed with the addition, the scalar multiplication, the Lie bracket, and the multiplication. This convention is very convenient, although somewhat misleading, because |$0$| is the unit of the multiplication law, the multiplication is not distributive with respect to the addition, and we have |$x^n=nx$| for all |$n\in\mathbf{Z}$|⁠. Let us write the Baker–Campbell–Hausdorff formula as \[xy=\sum_{i\ge 1}B_{i}(x,y)=(x+y)+\frac12[x,y]+\frac1{12}([x,[x,y]]-[y,[x,y]])+\dots,\] with |$B_i$| homogeneous of degree |$i$|⁠. This formula precisely makes sense in the pro-nilpotent completion of the free Lie |$\mathbf{Q}$|-algebra |$\mathfrak{f}$| on |$(x_1,x_2)$|⁠. Let |$m_i$| be the least common denominator of terms1 in |$B_i$|⁠. Small values are given by \[(m_1,m_2,\dots)=(1,2,12,24,720,1440,30240\dots).\] For |$i=2$|⁠, this means the stability under |$(x,y)\mapsto \frac12[x,y]$|⁠; in particular a strong subring is a Lie subring. In case the nilpotent Lie algebras, it also follows that a strong subring is closed under the group law defined by the Baker–Campbell–Hausdorff formula. There is an obvious notion of strong subring generated by a subset. (The definition of strong subring also makes sense in a |$c$|-step nilpotent Lie algebra over a ring of characteristic some power of any prime |$p>c$|⁠, because |$m_i$| is coprime to |$p$| for all |$i<p$|⁠.) We will need the following simple lemma. Lemma 4.1 is therefore a particular case of the following: Footnotes 1The definition of |$m_i$| is a bit sloppy, because “least common denominator of terms” refers to some choice of basis. It can be made more rigorous as follows: let |$\mathfrak{f}=\bigoplus_{i\ge 1}\mathfrak{f}_i$| be the standard grading of |$\mathfrak{f}$|⁠, so that |$B_i\in\mathfrak{f}_i$|⁠. Let |$\Lambda=\bigoplus_{i\ge 1}\Lambda_i$| be the Lie subring generated by |$(x_1,x_2)$|⁠, which is a free Lie |$\mathbf{Z}$|-algebra on |$(x_1,x_2)$|⁠. Here |$\Lambda_i$| is the additive subgroup generated by brackets of length |$i$| in |$x_1$| and |$x_2$|⁠. Define |$m_i$| as the smallest positive integer |$m$| such that |$B_i\in m^{-1}\Lambda_i$|⁠. Now if |$B_i$| is written with respect to a |$\mathbf{Z}$|-basis of |$\Lambda_i$|⁠, then |$m_i$| is indeed the smallest common denominator of the coefficients of |$B_i$|⁠, and indeed the Baker–Campbell–Hausdorff is usually written with respect to such a basis (namely a Hall basis, see [3, Section II.2.11]). References [1] Bass H. “The degree of polynomial growth of finitely generated nilpotent groups.” Proc. London Math. Soc. s3-25 , no. 4 ( 1972 ): 603 – 14 . [2] Borel A. , and Springer. T. “Rationality properties of linear algebraic groups II.” Tohoku Math. J. 20 ( 1968 ): 443 – 97 . Google Scholar Crossref Search ADS [3] Bourbaki N. “Groupes et algèbres de Lie, Algèbres de Lie libres, Groupes de Lie.” Chap. 2 and 3 in éléments de mathématique . Hermann, Paris : Actualités Scientifiques et Industrielles, 1972 . [4] Bourbaki N. “Algèbre.” In éléments de mathématique , chapters 4 and 7. Masson, 1981 . [5] Bou-Rabee K. and Cornulier. Y. “Systolic growth of linear groups.” Proc. Amer. Math. Soc. 144 , no. 2 ( 2016 ): 529 – 33 . Google Scholar Crossref Search ADS [6] Bou-Rabee K. and Studenmund. D. “Full residual finiteness growths of nilpotent groups.” Israel J. Math. 214 , no. 1 ( 2016 ): 209 – 33 . Google Scholar Crossref Search ADS [7] Breuillard E. “Geometry of groups of polynomial growth and shape of large balls.” Groups Geom. Dyn. 8 ( 2014 ): 1 – 64 . Google Scholar Crossref Search ADS [8] Corwin L. , and Greenleaf. F. Representation of Nilpotent Lie Groups and Their Applications . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1990 . [9] Cornulier Y. “Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups.” Bull. Soc. Math. France 144 , no. 4 ( 2016 ): 693 – 744 . Google Scholar Crossref Search ADS [10] de Graaf W. “Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2.” J. Algebra 309 ( 2007 ): 640 – 53 . Google Scholar Crossref Search ADS [11] Gromov M. “Groups of polynomial growth and expanding maps.” Inst. Hautes Études Sci. Publ. Math. 53 ( 1981 ): 53 – 73 . Google Scholar Crossref Search ADS [12] Gromov M. “Systoles and intersystolic inequalities.” In Actes de la table ronde de géométrie différentielle, 291 – 362 . Séminaires et Congrès 1, Soc. Math (Luminy, 1992) France, Paris, 1996 . [13] Guivarch Y. “Croissance polynomiale et périodes des fonctions harmoniques.” Bull. Soc. Math. France 101 ( 1973 ): 333 – 79 . Google Scholar Crossref Search ADS [14] Jenkins J. “Growth of connected locally compact groups.” J. Funct. Anal. 12 ( 1973 ): 113 – 27 . Google Scholar Crossref Search ADS [15] Losert V. “On the structure of groups with polynomial growth.” Math. Z. 195 ( 1987 ): 109 – 17 . Google Scholar Crossref Search ADS [16] Magnin L. “Determination of 7-dimensional indecomposable nilpotent complex Lie algebras by adjoining a derivation to 6-dimensional Lie algebras.” Algebr. Represent. Theory 13 , no. 6 ( 2010 ): 723 – 53 . Google Scholar Crossref Search ADS [17] Pansu P. “Croissance des boules et des géodésiques fermées dans les nilvariétés.” Ergodic Theory Dynam. Systems 3 , no. 3 ( 1983 ): 415 – 45 . Google Scholar Crossref Search ADS [18] Segal D. Polycyclic Groups , vol. 82. Cambridge Tracts in Mathematics. Cambridge : Cambridge University Press, 1983 . © The Author 2017. Published by Oxford University Press. All rights reserved. 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Claeys,, Tom;Girotti,, Manuela;Stivigny,, Dries

doi: 10.1093/imrn/rnx202pmid: N/A

Abstract We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit distributions can be expressed as Fredholm determinants of integral operators associated to kernels built out of Meijer |$G$|-functions or Wright’s generalized Bessel functions. They generalize in a natural way the hard edge Bessel kernel Fredholm determinant. We express the logarithmic derivatives of the Fredholm determinants identically in terms of a |$2\times 2$| Riemann–Hilbert problem, and use this representation to obtain the so-called large gap asymptotics. 1 Introduction It is well known that gap probabilities and extreme eigenvalue distributions of random matrices whose eigenvalues follow a determinantal point process can be expressed as Fredholm determinants corresponding to integral kernel operators. As the size of the random matrices tends to infinity, universal limit distributions arise, depending on the scaling regime. Three classical limit distributions for Hermitian random matrices are the Fredholm determinants associated to the sine kernel, the Airy kernel, and the Bessel kernel. Loosely speaking, the sine kernel determinant describes gap probabilities in the bulk of the spectrum, the Airy kernel determinant describes extreme eigenvalue distributions near soft edges, and the Bessel kernel determinant describes extreme eigenvalue distributions near certain hard edges. Those three determinants are well understood. In particular, they can be expressed identically in terms of Painlevé transcendents [30, 42–44] and their asymptotic behaviour for large gaps is known: see [16, 17, 25, 32, 45] for the sine, [3, 15, 43] for the Airy, and [18, 26] for the Bessel asymptotics. Recently, there has been a lot of interest in Wishart-type products of random matrices [1, 2, 31, 35–37] and in Muttalib–Borodin ensembles [9–11, 13, 27, 34, 38, 47, 48]. Whereas the local microscopic behaviour of the particles in the bulk and at the soft edge of the spectrum in these models is governed by the usual sine kernel and Airy kernel laws [37, 47], new universal limiting kernels near the hard edge have been discovered in this context, associated to kernels built out of Meijer |$G$|-functions [36] and Wright’s generalized Bessel functions [10] (see (1.11)–(1.13) below). The study of the associated Fredholm determinants, which describe the limit distributions of the smallest eigenvalue, was initiated recently in [41] for products of random matrices and in [48] for Muttalib–Borodin ensembles, and remarkable systems of differential equations have been obtained. We contribute to these developments by obtaining large gap asymptotics, and by expressing the Fredholm determinants identically in terms of a |$2\times 2$| Riemann–Hilbert (RH) problem. The random matrix models. We are interested in three different types of random matrices, which we describe below. (1) Our first case of interest consists of random matrices |$M^{(1)}$| of the form \begin{equation} M^{(1)}=\left(G_r\ldots G_{2}G_1\right)^*G_r\ldots G_{2}G_1, \end{equation} (1.1) where each factor |$G_j$| is an independent complex Ginibre matrix of size |$(n+\nu_j)\times(n+\nu_{j-1})$|⁠, with |$\nu_0=0$|⁠, and |$r\in\mathbb N$|⁠, |$\nu_1,\ldots, \nu_r\in\mathbb N\cup\{0\}$|⁠. This means that all entries of |$G_j$| are independent complex standard Gaussians. The notation |$^*$| stands for the Hermitian conjugate. (2) Our second model consists of products of truncations of Haar distributed unitary matrices. We let |$M^{(2)}$| be of the form \begin{equation} M^{(2)}=\left(T_r\ldots T_{2}T_1\right)^*T_r\ldots T_{2}T_1, \end{equation} (1.2) where |$r\in\mathbb N$| and |$T_j$| is the upper left |$(n+\nu_j)\times(n+\nu_{j-1})$| truncation of a Haar distributed unitary matrix |$U_j$| of size |$\ell_j \times \ell_j$|⁠. We assume that |$U_1,\ldots, U_r$| are independent, that |$\nu_0 = 0$|⁠, |$\nu_1, ..., \nu_r\in\mathbb N\cup\{0\}$|⁠, and that |$\ell_j \geq n+\nu_j+1$| for |$j = 1, ..., r$|⁠. Moreover, we assume that |$\sum_{j=1}^r(\ell_j-n-\nu_j)\geq n$|⁠. (3) Finally, we consider random matrices |$M^{(3)}$| whose eigenvalue joint probability distributions take the form \begin{equation}\label{MB} \frac{1}{Z_n}\Delta(x_1,\ldots, x_n)\Delta(x_1^\theta,\ldots, x_n^\theta)\prod_{k=1}^n x_k^\alpha {\rm e}^{-nx_k}{\rm d} x_k, \end{equation} (1.3) with |$x_1,\ldots, x_n>0$|⁠, where |$\alpha>-1$|⁠, |$\theta>0$|⁠, and \begin{equation} \Delta(x_1,\ldots, x_n)=\prod_{1\leq i<k\leq n}(x_k-x_i). \end{equation} (1.4) Such densities are known as Muttalib–Borodin Laguerre ensembles and were shown recently to arise naturally as joint probability densities for the squared singular values of certain upper triangular matrix ensembles [11, 27]. The simplest case of models (1) and (3) is the Wishart/Laguerre ensemble. If we set |$r=1$| in (1), the matrix |$M^{(1)}$| has the form |$G^*G$| with |$G$| a complex Ginibre matrix of size |$(n+\nu_1)\times n$|⁠. Such a matrix is called a Wishart/Laguerre random matrix, and the joint probability distribution of the eigenvalues is given by (1.3) in the case |$\theta=1$| and |$\alpha=\nu_1$|⁠. The simplest case of model (2), corresponding to |$r=1$|⁠, is the Jacobi Unitary Ensemble (see [31, Section 4]). In each of the above models, the joint probability density function for the eigenvalues is a determinantal point process. The associated correlation kernels |$K_n^{(j)}$| for the eigenvalues of |$M^{(j)}$| have various representations; for us it is convenient that they can be expressed as double contour integrals of the form \begin{equation}\label{correlation kernel Kn intro} K_n^{(j)}(x,y)=\frac{1}{4\pi^2}\int_{\gamma}{\rm d} u\int_{\tilde\gamma}{\rm d} v\, \frac{F_n^{(j)}(u)}{F_n^{(j)}(v)}\frac{x^{-u}y^{v-1}}{u-v}, \qquad j=1,2,3 \end{equation} (1.5) for some functions |$F_n^{(j)}$| which depend on |$j$| and on the parameters |$\{\nu_j\}, \{\mu_k\}, \alpha, \theta$| in the model, and for some contours |$\gamma$| and |$\tilde\gamma$|⁠. Such expressions are known for each of the models corresponding to |$j=1,2,3$|⁠. For convenience of the reader, we describe the explicit form of |$F_n^{(j)}$| and of the shape of the contours |$\gamma$| and |$\tilde\gamma$| in Appendix, see Proposition A.1, although we will not use this in what follows. In cases (1) and (3), we will be interested in the limit where |$n\to +\infty$| for fixed values of the parameters |$\nu_1,\ldots, \nu_r$| and |$\alpha, \theta$|⁠. In case (2), if we let |$n\to + \infty$|⁠, we also need to let |$\ell_1, ..., \ell_r$| go to infinity. For each |$j$|⁠, we may choose either to let |$\ell_j-n$| go to infinity, or to keep |$\ell_j-n$| fixed. The large |$n$| behaviour of the eigenvalues of |$M^{(2)}$| will depend on these choices. We take |$J \subseteq \{2,...,r\}$| a subset of indices with cardinality |$0 \leq q := |J| < r$|⁠, and we let |$\ell_1,...,\ell_r$| go to infinity in such a way that \begin{align} & \ell_k - n \rightarrow +\infty, &\mbox{if}~ k \notin J,\\ \end{align} (1.6) \begin{align} &\ell_{k_j} - n = \mu_j>\nu_j, \, \nu_j\in\mathbb N\cup\{0\}, &\mbox{if}~k_j \in J. \end{align} (1.7) Smallest eigenvalue distribution. The smallest particle |$x^*:=\min_{1\leq m\leq n}x_m$| in the models which we study has a distribution given by \begin{equation} {\rm Prob}\left(x^*>s\right)=1+\sum_{m=1}^n \frac{(-1)^m}{m!}\int_{[0,s]^m}\det\left(K_n^{(j)}(x_i,x_\ell)\right)_{i,\ell=1}^m {\rm d} x_1\ldots {\rm d} x_m. \end{equation} (1.8) Since |$K_n^{(j)}$| is of rank |$n$|⁠, this is the standard series expansion of the Fredholm determinant of the integral operator acting on |$L^2(0,s)$| with kernel |$K_n^{(j)}$|⁠. We denote this Fredholm determinant by |$\det\left(1-\left.K_n^{(j)}\right|_{[0,s]}\right)$|⁠. Near the origin, the eigenvalue correlation kernels |$K_n^{(j)}$| admit scaling limits of the following form: for |$x,y>0$|⁠, we have \begin{equation}\label{scaling limit} \lim_{n\to +\infty}\frac{1}{c_n^{(j)}}K_n^{(j)}\left(\frac{x}{c_n^{(j)}},\frac{y}{c_n^{(j)}}\right)=\mathbb K^{(j)}(x,y),\qquad j=1,2,3, \end{equation} (1.9) where \begin{equation}\label{def cj} c_n^{(1)}=n,\qquad c_n^{(2)} = n\prod_{k\notin J} (\ell_k-n),\qquad c_n^{(3)}=n^{\frac{1}{\theta}}, \end{equation} (1.10) for some limiting kernels |$\mathbb K^{(j)}$|⁠. This was proved in [36] for |$j=1$|⁠, in [31] for |$j=2$|⁠, and in [10] for |$j=3$|⁠. The limiting kernels can be expressed in terms of Meijer |$G$|-functions for |$j=1,2$| and in terms of Wright’s generalized Bessel functions for |$j=3$|⁠. In particular, for the cases |$j=1, 2$| the kernel has the following expression (see [36, Formula 5.7] and [31, Formula 2.36]) \begin{gather}\label{MeijerGkernelex} \mathbb{K}^{(1)} (x,y) = \int_0^1 G^{1,0}_{0,r+1}\left(\left.\begin{array}{c}-\\ -\nu_0,-\nu_1, \dots,-\nu_r\end{array}\right\vert ux\right) G^{1,0}_{0,r+1}\left(\left.\begin{array}{c}-\\ \nu_1,\nu_2, \dots,\nu_r\nu_0\end{array}\right\vert uy\right) {\rm d} u \end{gather} (1.11) and \begin{gather} \mathbb{K}^{(2)} (x,y) = - \int_0^1 G^{1,q}_{q,r+1}\left(\left. \begin{array}{c} -\mu_1, \dots, -\mu_q \\ -\nu_0,-\nu_1, \dots,-\nu_r \end{array}\right\vert ux\right) G^{r,0}_{q,r+1}\left(\left. \begin{array}{c} -\mu_1, \dots, -\mu_q \\ -\nu_1,-\nu_2, \dots,-\nu_r,-\nu_0 \end{array}\right\vert uy\right) {\rm d} u, \end{gather} (1.12) while for the case |$j=3$| the kernel reads (see [10]) \begin{gather}\label{MeijerGkernelex3} \mathbb{K}^{(3)} (x,y) = \theta \int_0^1 J_{\frac{\alpha+1}{\theta}, \frac{1}{\theta}}(xu) \, J_{\alpha +1, \theta} \left((yu)^\theta\right) u^\alpha {\rm d} u. \end{gather} (1.13) In the special case |$\theta = \frac{m}{n} \in \mathbb{Q}$|⁠, the Wright’s generalized Bessel functions |$J_{\frac{n(\alpha+1)}{m}, \frac{n}{m}}$||$J_{\alpha+1, \frac{m}{n}}$| can be expressed as Meijer |$G$|-functions as well (see [48]). Furthermore, if |$\theta \in \mathbb N$| or |$\frac{1}{\theta} \in \mathbb N$|⁠, then the limiting kernels |$\mathbb K^{(3)}$| and |$\mathbb K^{(1)}$| are related, as it was shown in [35]. For us, it is important to express the limiting kernels as double contour integrals. As we will show in Proposition A.2, they can be expressed as \begin{equation} \label{limiting kernels1}\mathbb K^{(j)}(x,y)=\frac{1}{4\pi^2}\int_\gamma{\rm d} u\int_{\tilde\gamma}{\rm d} v\frac{F^{(j)}(u)}{F^{(j)}(v)}\frac{x^{-u}y^{v-1}}{u-v} ,\qquad j=1,2,3, \end{equation} (1.14) with \begin{align} &F^{(1)}(z)=\frac{\Gamma(z)}{\prod_{j=1}^r \Gamma\left(1+\nu_j -z\right)},\label{f1}\\ \end{align} (1.15) \begin{align} &F^{(2)}(z)=\frac{\Gamma(z)\prod_{k = 1}^q \Gamma\left(1+\mu_k -z\right)}{\prod_{j=1}^r \Gamma\left(1+\nu_j -z\right)} \quad (q= |J|),\label{f2}\\ \end{align} (1.16) \begin{align} &F^{(3)}(z)=\frac{\Gamma(z+\frac{\alpha}{2})}{\Gamma\left(\frac{\frac{\alpha}{2}+1-z}{\theta}\right)}.\label{f3} \end{align} (1.17) The contours |$\gamma$| and |$\tilde\gamma$| are such that |$\gamma$| lies to the right of the poles of |$F^{(j)}$| and |$\tilde\gamma$| lies to the left of the zeros of |$F^{(j)}$|⁠, such that the vertical line through |$\frac{1}{2}$| lies in between |$\gamma$| and |$\tilde\gamma$| (this is always possible, since |$\mu_k> \nu_k, \, \nu_k\in\mathbb N\cup\{0\}$| and |$\alpha>-1$|⁠), and such that |$\gamma$| and |$\tilde\gamma$| do not intersect. Both contours are oriented upwards, and they tend to infinity in sectors lying strictly in the left half plane (for |$\gamma$|⁠) or the right half plane (for |$\tilde\gamma$|⁠). The contours are illustrated in Figure 1 for |$j=3$|⁠, |$\alpha>0$| and |$\theta>0$|⁠. Fig. 1. View largeDownload slide The contours |$\gamma, \tilde\gamma$| involved in the double-integral representation of the kernel |$\mathbb{K}^{(3)}$|⁠, in the case |$\alpha>0$|⁠, for |$\theta>0$|⁠; the dots represent the poles and the zeroes of the function |$F^{(3)}$|⁠. The real value |$\frac{1}{2}$| lies always in between |$\gamma$| and |$\tilde\gamma$|⁠. Fig. 1. View largeDownload slide The contours |$\gamma, \tilde\gamma$| involved in the double-integral representation of the kernel |$\mathbb{K}^{(3)}$|⁠, in the case |$\alpha>0$|⁠, for |$\theta>0$|⁠; the dots represent the poles and the zeroes of the function |$F^{(3)}$|⁠. The real value |$\frac{1}{2}$| lies always in between |$\gamma$| and |$\tilde\gamma$|⁠. It is worth noting that the kernels |$\mathbb K^{(1)}$| also appear in Cauchy multi-matrix models [5, 7, 8]. The simplest case of all three limiting kernels is the same: if |$r=1$|⁠, |$\nu_1=\alpha$| in case |$j=1$|⁠, or if |$r=1$|⁠, |$q=0$|⁠, |$\nu_1=\alpha$| in case |$j=2$|⁠, or if |$\theta=1$| in case |$j=3$|⁠, we have \[F^{(1)}\left(z+ \frac{\alpha}{2}\right)=F^{(2)}\left(z +\frac{\alpha}{2}\right)=F^{(3)}(z)=\frac{\Gamma(z+\frac{\alpha}{2})}{\Gamma\left(\frac{\alpha}{2}+1-z\right)},\] and then the right-hand side of (1.14) is a well-known (see e.g., [47]) integral representation of the hard edge Bessel kernel. A slightly stronger version of (1.9) allows one to show that the large |$n$| limit of the smallest particle distribution is given by \begin{align}\label{lim eig distr Fredholm} \lim_{n\to +\infty}{\rm Prob}\left(c_n^{(j)}x^*>s\right)&=\lim_{n\to +\infty}\det\left(1-\left.K_n^{(j)}\right|_{[0,s/c_n^{(j)}]}\right) \nonumber \\ &= \det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)\!. \end{align} (1.18) We will justify the last equality in the above formula in Lemma A.3 and Corollary A.4, using standard results from [40, Theorem 2.21 and Addendum H] about trace class operators. The Fredholm determinants on the second line of (1.18) are the central objects in what follows. Fredholm determinants and RH problems. Let |$K$| be an integral operator acting on |$L^2(\Sigma)$|⁠, with |$\Sigma$| being a collection of oriented contours in the complex plane. We call the kernel |$K$| of the integrable form if \begin{equation*} K(u, v) = \frac{\textbf{f}(u)^T \textbf{g} (v)}{u - v}, \label{JSkernel2} \end{equation*} where |$\textbf{f}$| and |$\textbf{g}$| are |$p$|-dimensional column vectors of sufficiently smooth functions on |$\Sigma$|⁠, satisfying the condition |$\textbf{f}(u)^T \textbf{g}(u) =0$|⁠. Assume that the kernel |$K$| additionally depends on some deformation parameters |$\{\kappa_i\}$| and consider its Fredholm determinant |$\det (1 - K)$| as a function of such parameters: a well-known procedure due to Its-Izergin-Korepin-Slavnov [29] allows one to express the logarithmic derivatives |$\partial_{\kappa_i} \ln \det(1-K)$|⁠, for all |$ i$|⁠, in terms of a RH problem of size |$p\times p$|⁠. For |$p=2$|⁠, this RH representation is often useful to derive asymptotic properties of the Fredholm determinants by applying the Deift/Zhou steepest descent method on the RH problem. The RH method has led to rigorous large |$s$| asymptotics (which are large gap asymptotics for the underlying determinantal processes) for, among others, the sine, Airy, and Bessel kernel Fredholm determinants [3, 15, 16, 18, 33]. It is known [36, 41] that |$\mathbb K^{(1)}$| is of integrable form with |$p=r+1$|⁠, and that |$\mathbb K^{(3)}$| is of integrable form if |$\theta=a/b\in\mathbb Q$| with |$p$| depending on |$a$| and |$b$| [48]. Overall, except in the simplest case corresponding to the Bessel kernel, the associated RH problems are of size |$p\times p$| with |$p>2$| and it is not clear whether such RH problems can be used to obtain large |$s$| asymptotics. A key observation in this article (see Section 2), based on ideas from [6], is that the integral operators |$\left.\mathbb K^{(j)}\right|_{[0,s]}$| can be factorized in the form |$\mathcal{M}^{-1} \circ \mathbb H_s^{(j)} \circ \mathcal{M}$| for some suitable operator |$\mathcal M$|⁠, where |$\mathbb H_s^{(j)}$| is an integral operator with kernel of integrable form, with |$p=2$|⁠. Therefore, we have \[\det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)=\det\left(1-\mathbb H_s^{(j)}\right),\] and we can express |$\frac{{\rm d}}{{\rm d} s}\ln \det\left(1-\mathbb H_s^{(j)}\right)$| identically in terms of a |$2\times 2$| matrix RH problem. The fact that the RH problem is of size |$2\times 2$| is remarkable, and it is important to derive large |$s$| asymptotics. We now state the relevant |$2\times 2$| RH problem, with jump contours |$\gamma$| and |$\tilde\gamma$| as defined before. RH problem for |$Y$| (a) |$Y: \mathbb{C}\setminus(\gamma\cup\tilde\gamma)\to \mathbb{C}^{2\times 2}$| is analytic; (b) |$Y(z)$| has continuous boundary values |$Y_\pm(z)$| as |$z$| approaches the contour |$\gamma\cup\tilde\gamma$| from the left (⁠|$+$|⁠) and right (⁠|$-$|⁠), according to its orientation, and we have the jump relations \begin{equation}\label{RHP Y: jump} Y_+ (z)= Y_-(z) J^{(j)}(z),\qquad z \in \gamma \cup \tilde \gamma, \end{equation} (1.19) with jump matrix (the contours are as in Fig. 1) \begin{equation}\label{def Jj} J^{(j)}(z) = \begin{cases} \begin{pmatrix} 1&0\\ s^z F^{(j)}(z)^{-1}&1 \end{pmatrix},& z\in\tilde\gamma,\\ \begin{pmatrix} 1&- s^{-z} F^{(j)}(z) \\0&1 \end{pmatrix},& z\in\gamma, \end{cases} \end{equation} (1.20) where |$F^{(1)}$|⁠, |$F^{(2)}$|⁠, and |$F^{(3)}$| are as in (1.14); (c) as |$z\to\infty$|⁠, there exists a matrix |$Y_1=Y_1(s)$| such that \begin{equation}\label{as Y} Y = I + \frac{Y_1(s)}{z}+\mathcal{O}\left(\frac{1}{z^{2}}\right). \end{equation} (1.21) Using the Its–Izergin–Korepin–Slavnov procedure, we will show that a solution to this RH problem exists; uniqueness of the solution can be shown using standard techniques. The RH solution |$Y$| depends on the value of |$j=1,2,3$| and also on |$s$| and on the values of the parameters in each of the models, but we will simply write |$Y$| for notational convenience. Statement of results. As our first result, we establish an identity which expresses the logarithmic |$s$|-derivative of the Fredholm determinant in terms of the RH solution |$Y$|⁠. Theorem 1.1 (Differential identity for gap probabilities). Let |$\mathbb K^{(j)}$| be the kernels defined in (1.14) for |$j=1,2,3$|⁠, and let |$\det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)$| be the Fredholm determinant of the associated operators acting on |$[0,s]$| with |$s>0$|⁠. Then, we have the identity \begin{gather}\label{diff id} \frac{{\rm d}}{{\rm d} s} \ln \det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right) =- \frac{1}{s} \left(Y_{1}(s)\right)_{2,2}, \end{gather} (1.22) with |$\left(Y_{1}(s)\right)_{2,2} $| the |$(2,2)$|-entry of |$Y_1(s)$|⁠, and |$Y_1(s)$| defined by (1.21) in terms of the unique solution to the RH problem for |$Y$|⁠. □ We prove the above result in Section 2. This RH representation for the Fredholm determinant is particularly useful to study large |$s$| asymptotics. We will obtain large |$s$| asymptotics for |$Y$| using the Deift/Zhou steepest descent method and this will enable us to prove the following result. Theorem 1.2 (Large gap asymptotics). Let |$\mathbb K^{(j)}$| be the kernels defined in (1.14) for |$j=1,2,3$|⁠, and let |$\det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)$| be the Fredholm determinant of the associated operators acting on |$[0,s]$| with |$s>0$|⁠. For |$j=1,2,3$|⁠, there exist constants |$C^{(j)}$|⁠, |$c^{(j)}$| such that, as |$s \to +\infty$|⁠, \begin{equation} \det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)= C^{(j)}e^{- a^{(j)}s^{2\rho^{(j)}}+ b^{(j)}s^{\rho^{(j)}} +c^{(j)}\ln s} \left(1 + o(1)\right). \label{asymptoticstheta} \end{equation} (1.23) Here, the values of |$\rho^{(j)}$| are given by \begin{equation} \label{def rho} \rho^{(1)}=\frac{1}{r+1},\quad \rho^{(2)}=\frac{1}{r-q+1},\quad \rho^{(3)}=\frac{\theta}{\theta+1}, \end{equation} (1.24) the values of |$a^{(j)}$| by \begin{align} &a^{(1)}= \frac{r^{\frac{1-r}{1+r}} (r+1)^2}{4} \\ \end{align} (1.25) \begin{align} &a^{(2)}= \frac{(r-q)^{\frac{1-r+q}{1+r-q}} (r-q+1)^2}{4}\\ \end{align} (1.26) \begin{align} &a^{(3)}= \frac{\theta^{\frac{1-3\theta}{1+\theta}} (1+\theta)^2}{4}, \end{align} (1.27) and the values of |$b^{(j)}$| by \begin{align} & b^{(1)} = (r + 1) r^{-\frac{r}{r + 1}} \sum_{j = 1}^r \nu_j \\ \end{align} (1.28) \begin{align} & b^{(2)} = (r -q+ 1) (r-q)^{-\frac{r-q}{r-q + 1}} \left[\sum_{j = 1}^r \nu_j-\sum_{k = 1}^q \mu_k \right] \\ \end{align} (1.29) \begin{align} & b^{(3)} = \frac{\theta+1}{2} \theta^{-\frac{2\theta}{\theta+1}} \left[1+2\alpha-\theta\right], \end{align} (1.30) with |$\nu_{\min}=\min \{\nu_1, \ldots, \nu_r\}$|⁠. □ Remark 1.3 We are not able to evaluate the multiplicative constants |$C^{(j)}$| explicitly, since they arise as integration constants after integration of the logarithmic derivative (1.22). The evaluation of such constants is in general a hard task [33], and our method does not allow this. The constants |$c^{(j)}$| on the other hand can be computed in principle, but their expressions are involved. We comment on this in Subsection 4.3. □ Remark 1.4 As stated before, in the case |$\theta=1$|⁠, the kernel |$\mathbb K^{(3)}$| reduces to the hard edge Bessel kernel up to a re-scaling: \begin{equation} \mathbb K^{(3)}_{\alpha,\theta=1}(x, y) = 4 \mathbb K_{\operatorname{Bessel},\alpha}(4x,4y), \label{K3andKBessel} \end{equation} (1.31) with \begin{equation} \mathbb K_{\operatorname{Bessel},\alpha}(x,y)= \frac{J_{\alpha}(\sqrt{x}) \sqrt{y} J_{\alpha}'(\sqrt{y}) - J_{\alpha}(\sqrt{y}) \sqrt{x} J_{\alpha}'(\sqrt{x})}{2 (x-y)}. \end{equation} (1.32) Similarly, if |$r=1$|⁠, |$\nu_1=\alpha\in\mathbb N\cup\{0\}$|⁠, and |$q=0$|⁠, we have \begin{equation} \mathbb K^{(1)}_{r=1,\nu_1=\alpha}(x, y) = \mathbb K^{(2)}_{q=0, r=1, \nu_1=\alpha}(x, y)= 4 \left(\frac{y}{x}\right)^{\alpha/2} \mathbb K_{\operatorname{Bessel},\alpha}(4x,4y). \end{equation} (1.33) Therefore, in these cases, we have \begin{gather} \det\left(1 - \mathbb{K}^{(j)}\bigg|_{[0, s]}\right) = \det\left(1 - \mathbb{K}_{\text{Bessel}}\bigg|_{[0, 4s]}\right). \end{gather} (1.34) We then have \[\rho^{(j)}=\frac{1}{2},\quad a^{(j)}=1, \quad b^{(j)}=2\alpha,\] and we obtain \[\det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)= C^{(j)}{\rm e}^{-s+ 2\alpha \sqrt{s} +c^{(j)}\ln s} \left(1 + o(1)\right),\qquad s\to +\infty,\] which is consistent with [18, formula (9)] and [26, formula (5)]. □ Remark 1.5. In the case of the product of two Ginibre matrices (⁠|$j=1$| and |$r=2$|⁠), we have \[\rho^{(1)}=\frac{1}{3},\quad a^{(1)}=\frac{9}{2^{7/3}},\] and thus \begin{equation*} \det\left(1-\left.\mathbb K^{(1)}\right|_{[0,s]}\right)= C^{(1)}{\rm e}^{-\frac{9}{2^{7/3}}s^{\frac{2}{3}}+ b^{(1)} s^{\frac{1}{3}} +c^{(1)}\ln s} \left(1 + o(1)\right),\qquad s\to +\infty, \end{equation*} which is consistent with a recent result obtained by Witte and Forrester [46, Corollary 3.1]. Furthermore, in the special case |$\nu_1= -\frac{1}{2}$| and |$\nu_2=0$|⁠, the sub-leading coefficient is equal to |$b^{(1)}=-\frac{3}{2^{5/3}}$|⁠, which agrees with formula (3.82) from the same article [46]. The same agreement is achieved for |$j=3$|⁠, |$\alpha=0$|⁠, |$\theta=2$| after the identification |$s\mapsto 2\sqrt{s}$| (see [46, formula (3.79)]), the coefficients being |$a^{(3)}=\frac{9}{2^{11/3}}$| and |$b^{(3)}=-\frac{3}{2^{7/3}}$|⁠. □ Outline. The rest of the article is organized as follows. In Section 2, we will use the approach developed by Its-Izergin-Korepin-Slavnov [29] together with ideas from [6] to prove the differential identity in Theorem 1.1. In Section 3, we then apply the Deift/Zhou steepest descent method to the RH problem for |$Y$| to obtain large |$s$| asymptotics. This will allow us to prove Theorem 1.2 in Section 4. 2 Differential Identity in Terms of RH Problem 2.1 Some considerations on the limiting kernels We note first that [39, formula (5.11.9)] for |$x,y\in \mathbb{R}$|⁠, \begin{equation} \Gamma(x+iy)\sim\sqrt{2\pi}|y|^{x-\frac{1}{2}}{\rm e}^{-\frac{\pi|y|}{2}},\qquad y\to \pm\infty, \end{equation} (2.1) which implies that, as |$y\to\pm\infty$|⁠, uniformly in |$x$|⁠, \begin{align} &\label{as F1}F^{(1)}(x+iy)=\left|y\right|^{(r+1)\left(x-\frac{1}{2}\right)} {\rm e}^{\frac{\pi(r-1)}{2}\left|y\right|}\left|y\right|^{-\sum_{j=1}^r\nu_j}\mathcal O(1) ,\\ \end{align} (2.2) \begin{align} &\label{as F2}F^{(2)}(x+iy)=\left|y\right|^{(r-q+1)\left(x-\frac{1}{2}\right)} {\rm e}^{\frac{\pi(r-q-1)}{2}\left|y\right|}\left|y\right|^{\sum_{k=1}^q\mu_k-\sum_{j=1}^r\nu_j}\mathcal O(1) ,\\ \end{align} (2.3) \begin{align} &\label{as F3}F^{(3)}(x+iy)= |y|^{\left(1+\frac{1}{\theta}\right) x - \frac{1}{\theta}}{\rm e}^{\frac{\pi|y|}{2}\left(\frac{1}{\theta}-1 \right)}|y|^{\frac{\alpha}{2}\left(1-\frac{1}{\theta}\right)}\theta^{-\frac{x}{\theta}}\mathcal{O}(1). \end{align} (2.4) As a consequence, it is easily seen that the double integral in (1.14) is convergent for our choices of contours |$\gamma,\tilde\gamma$| (recall that |$\gamma$| lies to the left of the line |${\rm{Re}}~ z=1/2$|⁠, and that |$\tilde\gamma$| lies to the right of it). From the definition of the kernels |$\mathbb K^{(j)}$| in (1.14) and the functions |$F^{(j)}$| in (1.15)-(1.17), it follows that, for any choice of parameters |$\{\nu_j\}, \{\mu_j\}, \alpha, \theta$|⁠, there exists |$C,\delta>0$| such that for |$x,y$| sufficiently small, \begin{equation} \left|\mathbb K^{(j)}(x,y)\right|\leq C |xy|^{-\frac{1}{2}+\delta}. \end{equation} (2.5) This implies that, for |$j=1,2,3$| and |$s>0$|⁠, \begin{equation}\label{norm operators lim} \int_0^s\int_0^s \left|\mathbb K^{(j)}(x,y)\right|^2 {\rm d} x{\rm d} y<\infty, \end{equation} (2.6) which means that the integral operators defined by \begin{equation}\label{def integral operator}\left.\mathbb K^{(j)}\right|_{[0,s]} f(y)=\int_{0}^s \mathbb K^{(j)}(x,y)f(x){\rm d} x \end{equation} (2.7) are bounded linear operators from |$L^2(0,s)$| to itself. 2.2 Conjugation with Mellin transform We will now use the Mellin transform to write the Fredholm determinants in a simpler form. We recall the definition of the Mellin operator and its inverse \begin{gather} \mathcal{M}[f](t) = \int_0^\infty x^{t-1} f(x) {\rm d} x, \quad \quad \mathcal{M}^{-1}[\varphi](x) = \frac{1}{2\pi i}\int_{\frac{1}{2}+i\mathbb{R}} x^{-t} \varphi(t){\rm d} t, \end{gather} (2.8) which are isometries between |$L^2(0,+\infty)$| and |$L^2\left(\frac{1}{2}+i\mathbb{R} \right)$|⁠. Proposition 2.1 Let |$s>0$|⁠. For |$j=1,2,3$|⁠, we have the identity \begin{equation}\label{identity K H} \det\left(1-\left.\mathbb K^{(j)}\right|_{[0,s]}\right)=\det\left(1-\mathbb H_s^{(j)}\right), \end{equation} (2.9) where |$\mathbb H_s^{(j)}$| is the integral operator acting on |$L^2(\tilde\gamma)$| with kernel \begin{gather}\label{kernel J} \mathbb H_s^{(j)}(v,z) = \int_\gamma \frac{{\rm d} u}{2\pi i} s^{z-u} \frac{F^{(j)}(u)}{F^{(j)}(v)(v-u)(z-u)} . \end{gather} (2.10) □ Proof. Consider the operator |$\left.\mathbb K^{(j)}\right|_{[0,s]}$| acting on |$L^2(0,s)$| defined by (2.7). As an operator on |$L^2(0,+\infty)$|⁠, its kernel is given by |$\mathbb K^{(j)}(x,y)\chi_{[0,s]}(x)$|⁠, where |$\chi_{[0,s]}$| is the characteristic function of the interval |$[0,s]$|⁠. As a function of |$z$|⁠, the function \[k(z;s,y):=\int_{\tilde \gamma} \frac{{\rm d} v}{2\pi i} \int_\gamma \frac{{\rm d} u}{2\pi i} \frac{s^{-u}F^{(j)}(u)}{F^{(j)}(v) (v-u)(z-u)} y^{v -1}\] is analytic for |$z$| in the region at the right of |$\gamma$| and for sufficiently large |$z$|⁠, it can be bounded in absolute value by |$C/|z|$| for some |$C>0$|⁠. For |$s\neq x\in \mathbb{R}_+$|⁠, we now evaluate the improper integral \begin{equation}\int_{\frac{1}{2}+i\mathbb{R}}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z k(z;s,y):=\lim_{R\to +\infty}\int_{\frac{1}{2}-iR}^{\frac{1}{2}+iR}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z k(z;s,y).\label{integralk}\end{equation} (2.11) First, if |$x>s$|⁠, by analyticity we can deform the integration contour |$\left[\frac{1}{2}-iR,\frac{1}{2}+iR\right]$| into a semi-circle in the half plane |${\rm{Re}}~ z>\frac{1}{2}$| with radius |$R$|⁠. For |$R$| sufficiently large, we obtain in this way, \[\left|\int_{\frac{1}{2}-iR}^{\frac{1}{2}+iR}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z k(z;s,y)\right|\leq \frac{C}{2\pi}\int_{-\pi/2}^{\pi/2}{\rm e}^{R\log\left(\frac{s}{x}\right)\cos\theta}{\rm d}\theta.\] Clearly, the upper bound tends to |$0$| as |$R\to +\infty$|⁠, thus \[\int_{\frac{1}{2}+i \mathbb{R}}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z k(z;s,y)=0.\] Next, if |$x<s$|⁠, we need to deform the integration contour to the half plane |${\rm{Re}}~ z< \frac{1}{2}$|⁠. For example we can deform |$\frac{1}{2}+i\mathbb{R}$| to a contour |$\Sigma$| consisting of two straight half-lines starting at |$z=\frac{1}{2}$| and extending in the left-half plane. On such a contour, the |$z$|-integral in (2.11) is absolutely convergent, and we can use Fubini’s theorem to move the |$z$|-integral inside the |$u$|- and |$v$|-integrals. Using the residue theorem, we then obtain by (1.14), \begin{align} &\int_{\frac{1}{2}+i \mathbb{R}}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z k(z;s,y) \\ \end{align} (2.12) \begin{align} &\quad= \int_{\tilde \gamma} \frac{{\rm d} v}{2\pi i} \int_\gamma \frac{{\rm d} u}{2\pi i} \frac{s^{-u}y^{v -1}F^{(j)}(u)}{F^{(j)}(v) (v-u)} \int_{\Sigma}\frac{{\rm d} z}{2\pi i}\left(\frac{s}{x}\right)^z\frac{1}{z-u} =\mathbb K^{(j)}(x,y). \end{align} (2.13) Thus, both for |$x>s$| and |$x<s$|⁠, we have \begin{gather} \mathbb K^{(j)}(x,y)\chi_{[0,s]}(x) =\int_{\frac{1}{2}+i\mathbb{R}} \frac{{\rm d} z}{2\pi i} x^{-z} \int_{\tilde \gamma} \frac{{\rm d} v}{2\pi i} \int_\gamma \frac{{\rm d} u}{2\pi i} \frac{s^{z-u}F^{(j)}(u)y^{v -1}}{F^{(j)}(v) (v-u)(z-u)} . \end{gather} (2.14) Using this triple contour integral representation, we can show that, as an |$L^2(0,+\infty)$| operator, |$\left.\mathbb K^{(j)}\right|_{[0,s]}$| can be written as a Mellin conjugation of a simpler integral operator |$\mathbb H_s^{(j)}$|⁠: we have \begin{equation}{\mathbb K}^{(j)} \bigg|_{[0,s]} =\mathcal{M}^{-1} \circ \mathbb H_s^{(j)} \circ \mathcal{M},\label{conjugation} \end{equation} (2.15) where |$\mathbb H_s^{(j)}$| is the integral operator acting on |$L^2(\tilde\gamma)$| with kernel |$ \mathbb H_s^{(j)}(v,z)$| given by (2.10). This follows from the following computation on the level of the kernels: \begin{equation} \mathcal M\circ {\mathbb K}^{(j)} \bigg|_{[0,s]} (z,y)=s^zk(z;s,y) = \left(\mathbb H_s^{(j)} \circ \mathcal{M}\right)(z,y). \end{equation} (2.16) It follows from (2.15) that (2.9) holds, which completes the proof. ■ 2.3 Integrable form Proposition 2.2. Let |$s>0$|⁠. For |$j=1,2,3$|⁠, we have the identity \begin{gather}\det\left(1-\mathbb H_s^{(j)}\right)= \det \left(1-\mathbb M_s^{(j)} \right), \end{gather} (2.17) where |$\mathbb M_s^{(j)}$| is the integral operator acting on |$L^2(\gamma\cup\tilde\gamma)$| with kernel \begin{equation}\label{integrable form kernel} \mathbb M_s^{(j)}(u,v) = \frac{\textbf{f}(u)^T \textbf{g}(v)}{u-v} \end{equation} (2.18) where |$\textbf{f}$| and |$\textbf{g}$| are given by \begin{equation}\textbf{f}(u) = \frac{1}{2\pi i}\left[\begin{array}{c} \chi_{\gamma}(u) \\ s^u \chi_{\tilde \gamma} (u) \end{array}\right] , \quad \quad \textbf{g}(v) = \left[\begin{array}{c} -F^{(j)}(v)^{-1} \chi_{\tilde \gamma}(v) \\ s^{-v}F^{(j)}(v) \chi_\gamma(v) \end{array} \right], \end{equation} (2.19) with |$F^{(1)}$|⁠, |$F^{(2)}$|⁠, |$F^{(3)}$| defined by (1.15)–(1.17), and where |$\chi_{\gamma}$| (respectively, |$\chi_{\tilde \gamma}$|⁠) is the characteristic function of the contour |$\gamma$| (respectively, |$\tilde \gamma$|⁠). □ Proof. The operator |$\mathbb H_s^{(j)}$| on |$L^2(\tilde\gamma)$| can be written as the composition |$A^{(j)}\circ B^{(j)}$| of two operators |$A^{(j)}:L^2(\gamma)\to L^2(\tilde\gamma)$| and |$B^{(j)}:L^2(\tilde\gamma)\to L^2(\gamma)$|⁠, where |$A^{(j)}$| and |$B^{(j)}$| are integral operators with kernels \begin{equation} A^{(j)}(u,z) = \frac{s^{z-u} F^{(j)}(u)}{2\pi i (z-u)}, \quad \quad B^{(j)}(v,u) = \frac{1}{F^{(j)}(v)(v-u)}. \end{equation} (2.20) It is then straightforward to check that |$\int_{\tilde\gamma} \int_{\gamma} \left|A^{(j)}\left(\eta,\xi\right)\right|^2 |{\rm d} \eta| |{\rm d} \xi|<+\infty$| and hence |$A^{(j)}$| is Hilbert–Schmidt, and similarly for |$B^{(j)}$|⁠. Therefore, as composition of two Hilbert–Schmidt operators, |$\mathbb H_s^{(j)}$| is trace-class. We now prove that the operators |$A^{(j)}$| and |$B^{(j)}$| are themselves trace-class. We can write |$B^{(j)}=B_2\circ B_1$| as the composition of two Hilbert–Schmidt operators as follows: \begin{gather} L^2(\tilde \gamma) \xrightarrow{B_1} L^2\left({\scriptstyle\frac{1}{2}}+ i\mathbb{R} \right) \xrightarrow{B_2} L^2(\gamma) \nonumber \\ f(v) \mapsto \int_{\tilde \gamma} {\rm d} v \frac{f(v)}{F^{(j)}(v) (v-w)} \mapsto \int_{\frac{1}{2}+i\mathbb{R}} \frac{{\rm d} w}{2\pi i} \int_{\tilde \gamma}{\rm d} v\frac{f(v)}{F^{(j)}(v) (v-w)(u-w)}. \nonumber \end{gather} Similarly, we have |$A^{(j)}=A_2\circ A_1$| with \begin{gather} L^2(\gamma) \xrightarrow{A_1} L^2\left({\scriptstyle \frac{1}{2}}+ i\mathbb{R} \right) \xrightarrow{A_2} L^2(\tilde \gamma) \nonumber \\ g(u) \mapsto \int_{\gamma} \frac{{\rm d} u}{2\pi i} \frac{s^{-u}F^{(j)}(u)g(u)}{w-u} \mapsto \int_{\frac{1}{2}+i\mathbb{R}} \frac{{\rm d} w}{2\pi i} \int_{\gamma} \frac{{\rm d} u}{2\pi i} \frac{s^{z-u}F^{(j)}(u)g(u)}{(w-u)(z-w)}. \nonumber \end{gather} It is now easy to verify that the kernels of |$A_1, A_2$| and |$B_1, B_2$| are Hilbert–Schmidt, hence |$A^{(j)}$| and |$B^{(j)}$| are trace-class. As an operator acting on the Hilbert space |$L^2(\gamma) \oplus L^2(\tilde\gamma)$|⁠, we can write |$\mathbb M_s^{(j)}$| as a |$2\times 2$| matrix of operators, \[\mathbb M_s^{(j)}=\left[\begin{array}{c|c} 0&A^{(j)}\\\hline B^{(j)} &0 \end{array}\right].\] Moreover, we have the following identities, \begin{eqnarray*}\det \left(1-\mathbb H_s^{(j)}\right)&=&\det \left(1- \left[\begin{array}{c|c} A^{(j)}\circ B^{(j)}&0\\\hline 0&0 \end{array}\right]\right)\\&=&\det \left(1- \left[\begin{array}{c|c} A^{(j)}\circ B^{(j)}&0\\\hline B^{(j)} &0 \end{array}\right]\right)\\&=& \det \left(1 + \left[\begin{array}{c|c} 0& A^{(j)}\\\hline 0 &0 \end{array}\right]\right)\det \left(1 - \left[\begin{array}{c|c} 0&A^{(j)}\\\hline B^{(j)} &0 \end{array}\right]\right) \\&=& \det \left(1 - \mathbb{M}_s^{(j)} \right), \end{eqnarray*} and the result is proved. ■ Following the procedure developed by Itz et al. [29], one can relate an integral operator with kernel of the integrable form (2.18) to the RH problem for |$Y$| given in (1.19)–(1.21), where we note that the jump matrix |$J^{(j)}$| takes the form \[J^{(j)}(z) = I - 2\pi\ i \textbf{f}(z) \textbf{g}(z)^T . \] In particular, the resolvent of the operator |$\mathbb M_s^{(j)}$| exists if and only if the solution to the above RH problem exists as well, and logarithmic derivatives of the Fredholm determinant |$\det(1-\mathbb M_s^{(j)})$| with respect to deformation parameters can be expressed in terms of |$Y$|⁠. In our situation, |$s$| plays the role of the deformation parameter, and we can use the results from [4, Section 5.1] and [6] (proved for general deformation parameters) to conclude that \begin{gather} \frac{{\rm d}}{{\rm d} s} \ln \det \left(1-\mathbb M_s^{(j)}\right) = \int_{\gamma\cup\tilde\gamma} \operatorname{Tr}\left[Y_-^{-1}(z) Y_-'(z)\, \partial_s J (\lambda) J^{-1}(z) \, \right] \, \frac{{\rm d} z}{2\pi i} \label{Misomonodromictau} \end{gather} (2.21) where |$'$| refers to the partial derivative with respect to |$z$|⁠. In the original formula in [6], an additional term is present which depends exclusively on the jumps of the RH problem, but this term is identically zero in our case. Furthermore, a simple calculation shows that \begin{align} &\int_{\gamma\cup\tilde\gamma} \operatorname{Tr}\left[Y_-^{-1}(z) Y_-'(z)\, \partial_s J (z) J^{-1}(z) \, \right] \, \frac{{\rm d} z}{2\pi i} \nonumber \\ &\quad= \int_{\gamma\cup\tilde\gamma} \frac{z}{2s} \left(\operatorname{Tr}\left[Y^{-1}(z) Y'(z) \sigma_3 \, \right]_+ - \operatorname{Tr}\left[Y^{-1}(z) Y'(z) \sigma_3 \, \right]_-\right) \, \frac{{\rm d} z}{2\pi i}, \label{2.21} \end{align} (2.22) with $$\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ the third Pauli matrix. Now we use a contour deformation argument to simplify this expression. Using the asympotic behaviour (1.21) of |$Y(z)$| as |$z \rightarrow \infty$|⁠, one sees that the contribution to the integral (2.22) of the |$-$| side of |$\gamma$| cancels out with the one of the |$+$| side of |$\tilde\gamma$|⁠. Furthermore, the integrals over the |$+$| side of |$\gamma$| and the |$-$| side of |$\tilde\gamma$| can be deformed, and their contribution is equal to the integral of a large clockwise oriented circle. In this way, we obtain that the above expression is equal to \begin{gather} = - \lim_{R\to\infty}\frac{1}{2s} \int_{C_R} \frac{\operatorname{Tr}\left[Y_1 \sigma_3 \, \right]}{z} \, \frac{{\rm d} z}{2\pi i} = - \frac{1}{s} \left(Y_{1}\right)_{2,2} , \end{gather} (2.23) where |$C_R$| is a counterclockwise oriented circle of radius |$R$| around |$0$|⁠. The last equality follows because |$Y_1$| is traceless, which in turn follows from the fact that |$\det Y (z) = 1$|⁠, for all |$z \in \mathbb{C} \setminus \left(\gamma \cup \tilde \gamma\right)$|⁠. This completes the proof of Theorem 1.1. Remark 2.3. The integral |$\int_{\gamma\cup\tilde\gamma} \operatorname{Tr}\left[Y_-^{-1} Y_-' \partial J J^{-1} \right] \frac{{\rm d} z}{2\pi i}$| can be interpreted in terms of the theory of isomonodromic |$\tau$|-functions [30], as explained in detail in [4]. □ Remark 2.4. The above procedure can be generalized to the case of several intervals instead of the single interval |$[0,s]$|⁠, similar to [6]. Let |$ J := \bigcup_{j=1}^{m} [a_{2j-1}, a_{2j}]$| be a collection of disjoint intervals on the positive real line: |$0 \leq a_1 < a_2 < \ldots < a_{2m}$|⁠. Then, \begin{gather} \mathbb{K}^{(j)}\bigg|_{J} = \sum_{k=1}^{2m} (-1)^k \mathbb{K}^{(j)}\bigg|_{[0,a_{k}]}, \end{gather} (2.24) and we can still write this operator as a conjugation with a Mellin transform as in (2.15). Then the adaption of the proofs of Propositions 2.1 and 2.2 is straightforward, and it leads to an integrable operator |$\mathbb M_s^{(j)}$| of the form (2.18), but now with |$\textbf{f}$| and |$\textbf{g}$| of larger dimension. This would lead to a RH problem of larger size, depending on the number of endpoints |$2m$|⁠. □ 3 Asymptotic Analysis of the RH Problem The aim of this section is to study the asymptotic behaviour as |$s\to +\infty$| of the solution |$Y=Y(z ;s)$| to the RH problem stated in Section 1. In particular, we will be interested in the |$(2,2)$|-entry of the matrix |$Y_1$|⁠, defined in (1.21), which appears in the logarithmic derivative of the Fredholm determinant in each of the cases |$j = 1, 2, 3$|⁠, as already proved in Section 2. To achieve the asymptotic results stated in Theorem 1.2, we will apply a series of invertible transformations to the RH problem for |$Y$| to obtain a RH problem for which we can easily inspect the large |$s$| asymptotics of the solution. This procedure is known as the Deift/Zhou steepest descent method [22, 23]. 3.1 First transformation |$Y \mapsto U$| In order to simplify the analysis of the RH problem, we define \begin{equation} \label{eq: transfYT} U(\zeta) := s^{\frac{\tau^{(j)}}{2}\sigma_3} Y\left(i s^{\rho^{(j)}} \zeta + \tau^{(j)} \right)s^{-\frac{\tau^{(j)}}{2}\sigma_3}, \end{equation} (3.1) with |$\tau^{(j)}$| depending on the parameters in the models corresponding to |$j = 1, 2, 3$| as follows, \begin{align*} & \tau^{(1)} = \frac{\nu_{\min} + 1}{2}, && \rho^{(1)} = \frac{1}{r + 1}, \\ & \tau^{(2)} = \frac{\nu_{\min} + 1}{2}, && \rho^{(2)} = \frac{1}{r - q + 1}, \\ & \tau^{(3)} = \frac{1}{2}, && \rho^{(3)} = \frac{\theta}{\theta + 1}, \end{align*} where we recall that |$\nu_{\min} := \min \{\nu_1, \ldots, \nu_r\}$| and |$q := |J|$|⁠. From now on, when there is no possible confusion, we omit the |$j$|-dependence in our notations, and we will simply write |$\tau, \rho, F$| instead of |$\tau^{(j)}, \rho^{(j)}, F^{(j)}$|⁠. The jump contours |$\gamma$| and |$\tilde\gamma$| are transformed into contours |$\gamma_U:=\{\zeta \in \mathbb{C} :is^{\rho^{(j)}}\zeta+\tau^{(j)}\in\gamma\}$| in the upper half plane and |$\tilde\gamma_U:=\{\zeta \in \mathbb{C} :is^{\rho^{(j)}}\zeta+\tau^{(j)}\in\tilde\gamma\}$| in the lower half plane, both oriented from left to right. The contours |$\gamma_U$| and |$\tilde\gamma_{U}$| depend on |$s$| and as |$s\to +\infty$| they both approach |$0$|⁠. The poles of the jump matrices |$J_U$| now lie on the imaginary axis and accumulate towards the origin in the large-|$s$| limit. |$U$| satisfies the following conditions. Fig. 2. View largeDownload slide The contours in the RH problem for |$U$|⁠. The poles appearing in the jump matrices lie now on the imaginary axis and accumulate towards the origin as |$s \rightarrow + \infty$|⁠. Fig. 2. View largeDownload slide The contours in the RH problem for |$U$|⁠. The poles appearing in the jump matrices lie now on the imaginary axis and accumulate towards the origin as |$s \rightarrow + \infty$|⁠. RH problem for |$\boldsymbol{U}$| (a) |$U$| is analytic in |$\mathbb{C} \setminus (\gamma_U\cup\tilde\gamma_U)$|⁠; (b) |$U_+(\zeta) = U_-(\zeta) J_U(\zeta)$| for |$\zeta\in\gamma_U\cup\tilde\gamma_U$| with \begin{equation}\label{def: JT1} J_U(\zeta) = \begin{cases}\begin{pmatrix}1&-s^{-i s^{\rho} \zeta} F\left(i s^{\rho} \zeta + \tau\right)\\0&1 \end{pmatrix} & \text{if}~ \zeta \in \gamma_U, \\ \begin{pmatrix}1&0 \\ s^{i s^{\rho} \zeta}F\left(i s^{\rho} \zeta + \tau\right)^{-1} &1 \end{pmatrix} & \text{if}~ \zeta \in \tilde{\gamma}_U;\end{cases} \end{equation} (3.2) (c) |$\displaystyle U(\zeta) = I + \frac{U_1(s)}{\zeta} + \mathcal{O}\left(\frac{1}{\zeta^2}\right)$| as |$\zeta \to \infty.$| On the other hand, from (3.1) and condition (c) in the RH problem for |$Y$|⁠, it follows that \begin{equation*} U(\zeta) = I + \frac{Y_1(s)}{i s^{\rho}} \zeta^{-1} + \mathcal{O}\left(\zeta^{-2}\right), \qquad \text{as} \zeta \to \infty \end{equation*} which implies the identity \begin{equation}\label{identity Fredholm U} \frac{{\rm d}}{{\rm d} s} \ln \det \left(1 - \mathbb{K}^{(j)}\bigg|_{[0,s]}\right) =- i s^{\rho^{(j)} - 1} \left(U_1(s)\right)_{2,2}. \end{equation} (3.3) 3.2 Deformation of the contours and transformation |$U\mapsto T$| By analyticity of the jump matrices |$J_U$|⁠, it follows that the RH solution |$U$| can be analytically continued from the region above |$\gamma_U$| to |$\mathbb{C}\setminus[0,-i\infty)$|⁠. We write |$U^{\rm I}$| for this analytic extension, which is defined as \begin{equation} U^{\rm I}(\zeta)=\begin{cases}U(\zeta),&\mbox{above}~ {\gamma_U,}\\ U(\zeta)\begin{pmatrix}1&-s^{-i s^{\rho} \zeta} F\left(i s^{\rho} \zeta + \tau\right)\\0&1 \end{pmatrix},&\mbox{below} ~{\gamma_U.} \end{cases} \end{equation} (3.4) Similarly, we can continue |$U$| from the region between |$\gamma_U$| and |$\tilde\gamma_U$| to |$ \mathbb{C}\setminus i\mathbb{R}$|⁠, and we denote this function by |$U^{\rm II}$|⁠. Finally, we can continue |$U$| from the region below |$\tilde\gamma_U$| to |$\mathbb{C}\setminus[0,+i\infty)$|⁠, and we denote this function by |$U^{\rm III}$|⁠. These analytic continuations can be used to deform |$\gamma_U$| and |$\tilde \gamma_U$|⁠. We will deform them in such a way that they partly coincide. To that end, we define contours |$\Sigma_1,\Sigma_2,\ldots, \Sigma_5$| and regions |${\rm I}, {\rm II},{\rm III}, {\rm IV}$| as shown in Figure 3, namely Fig. 3. View largeDownload slide The contour setting for the RH problem |$T(\zeta)$|⁠. Fig. 3. View largeDownload slide The contour setting for the RH problem |$T(\zeta)$|⁠. \begin{equation}\label{def Sigma} \Sigma_5=[b_1,0]\cup [0,b_2], \ \Sigma_2=-\overline{\Sigma_1}=b_2+{\rm e}^{i(\phi+\epsilon)}(0,+\infty),\ \Sigma_4=-\overline{\Sigma_3}=b_2+{\rm e}^{-i\epsilon}(0,+\infty), \end{equation} (3.5) with |$0<\epsilon<\pi/10$|⁠. The endpoints |$b_1 = b_1^{(j)}$| and |$b_2=b_2^{(j)}$| are such that |$b_2=-\overline{b_1}=b{\rm e}^{i\phi}$| and will be determined later. This type of contour will later turn out to be suitable in the steepest descent analysis for case |$j=1$|⁠, |$j=2$|⁠, and |$j=3$| with |$\theta \leq 1$|⁠. In what follows, we focus on these cases. If |$j=3$| and |$\theta > 1$|⁠, we need to deform the contours in a different way. We will comment on the changes which have to be made in this case in Remark 3.2. We define \begin{equation}\label{RHPforT} T(\zeta)=\begin{cases}U^{\rm I}(\zeta),&\mbox{in region I,}\\ U^{\rm II}(\zeta),&\mbox{in region II and IV,}\\ U^{\rm III}(\zeta),&\mbox{in region III.} \end{cases} \end{equation} (3.6) It is straightforward to check that the jump matrices for |$T$| on |$\Sigma_1,\ldots, \Sigma_4$| are the same as the ones for |$U$| on the corresponding contours |$\gamma_U$| and |$\tilde\gamma_U$|⁠. On |$\Sigma_5$|⁠, the jump matrix for |$T$| is obtained by multiplying the jump matrix |$J_U$| in (3.2) on |$\tilde\gamma$| with the one on |$\gamma$|⁠. RH problem for |$T$| (a) |$T$| is analytic in |$\mathbb{C} \setminus (\Sigma_1\cup\ldots\cup\Sigma_5)$|⁠; (b) |$T_+(\zeta) = T_-(\zeta) J_T(\zeta)$| for |$\zeta\in\Sigma_1\cup\ldots\cup\Sigma_5$| with \begin{equation}\label{def: JT1-second} J_T(\zeta) = \begin{cases}\begin{pmatrix}1&-s^{-i s^{\rho} \zeta} F\left(i s^{\rho} \zeta + \tau\right)\\0&1 \end{pmatrix} & \text{if}~ \zeta \in \Sigma_1\cup\Sigma_2; \\ \begin{pmatrix}1&0 \\ s^{i s^{\rho} \zeta}F\left(i s^{\rho} \zeta + \tau\right)^{-1} &1 \end{pmatrix} & \text{if}~ \zeta \in \Sigma_3\cup\Sigma_4;\\ \begin{pmatrix}1&-s^{-i s^{\rho} \zeta} F\left(i s^{\rho} \zeta + \tau\right) \\ s^{i s^{\rho} \zeta}F\left(i s^{\rho} \zeta + \tau\right)^{-1} &0 \end{pmatrix} & \text{if}~ \zeta \in \Sigma_5;\end{cases} \end{equation} (3.7) (c) |$\displaystyle T(\zeta) = I + \frac{T_1(s)}{\zeta} + \mathcal{O}\left(\frac{1}{\zeta^2}\right)$| as |$\zeta \to \infty,$| with |$T_1(s)=U_1(s)$|⁠. Before proceeding with the next transformation, we will rewrite the jump matrix |$J_T$|⁠. Recalling Stirling’s approximation formula for |$z \to \infty$| and |$|\text{arg} z| < \pi$|⁠, \begin{equation} \ln \Gamma (z) = z \ln z - z - \frac{1}{2} \ln z + \frac{1}{2} \ln 2\pi + \frac{1}{12 z} + \mathcal{O}\left(\frac{1}{z^3}\right), \end{equation} (3.8) we obtain from (1.15)–(1.17) that as |$s \to + \infty$| with |$\zeta$| not too close to |$0$| such that |$s^{\rho} \zeta \to \infty$|⁠, we have \begin{align} \ln F\left(i s^{\rho}\zeta + \tau \right) &= i s^{\rho} \ln (s) \zeta + is^{\rho}\left[c_1 \zeta \ln(i \zeta) + c_2 \zeta \ln(-i \zeta) + c_3 \zeta\right]\notag \\ &\quad + c_4 \ln(s)+ c_5 \ln(i \zeta) +c_6 \ln(-i \zeta) + c_7 + \frac{c_8}{i s^\rho\zeta} + \mathcal{O}\left(\frac{1}{s^{2\rho}\zeta^2}\right)\label{as F}, \end{align} (3.9) for some constants |$\left\{c_i =c_i^{(j)} \right\}_{i=1,\ldots, 8}$|⁠, depending on the parameters |$\{\nu_k\}, \{\mu_k\}, \alpha, \theta$|⁠, and with principal branches of the logarithms. We will use this for |$\zeta\in\Sigma_1\cup\ldots\cup\Sigma_5$| and |$s^\rho\zeta$| large. The constants |$\{c_i\}_{i = 1, \ldots, 7}$| are given by \begin{align} &j=1: && c_1 = 1, && c_2 = r, \nonumber \\ & && c_3 = -(r+1), && \displaystyle c_4 = \frac{\nu_{\text{min}}}{2} - \frac{1}{r+1} \sum_{k=1}^r \nu_k, \nonumber \\ & && \displaystyle c_5 = \frac{\nu_{\text{min}}}{2}, && \displaystyle c_6 = r\frac{\nu_{\text{min}}}{2} - \sum_{k=1}^r \nu_k, \nonumber \\ & && \displaystyle c_7 = \frac{1-r}{2}\ln 2\pi, && \displaystyle c_8 = \frac{r+1}{8}\left(\nu_{\rm min}^2-\frac{1}{3}\right) + \frac{1}{2} \sum_{k=1}^r \nu_k^2 - \frac{\nu_{\rm min}}{2}\sum_{k=1}^r \nu_k; \label{eq: ci_j1} \\ \end{align} (3.10) \begin{align} & j=2: && c_1 = 1, && c_2 = r-q, \nonumber \\ & && c_3 = -(r-q+1), && \displaystyle c_4 = \frac{\nu_{\text{min}}}{2} +\frac{1}{r-q+1}\left[\sum_{k=1}^q \mu_k - \sum_{k=1}^r \nu_k \right], \nonumber \\ & && \displaystyle c_5 = \frac{\nu_{\text{min}}}{2}, && \displaystyle c_6 = (r-q)\frac{\nu_{\text{min}}}{2} + \sum_{k=1}^q \mu_k - \sum_{k=1}^r \nu_k, \nonumber \\ & && \displaystyle c_7 = \frac{1+q-r}{2}\ln 2\pi, && c_8 = \displaystyle \frac{r-q+1}{8}\left(\nu_{\rm min}^2-\frac{1}{3}\right) + \frac{1}{2} \left(\sum_{k=1}^r \nu_k^2 - \sum_{k=1}^q \mu_k^2 \right) \nonumber \\ & && &&\qquad \displaystyle - \frac{\nu_{\rm min}}{2} \left(\sum_{k=1}^r \nu_k - \sum_{k=1}^q \mu_k \right); \label{eq: ci_j2} \\ \end{align} (3.11) \begin{align} & j=3: \ \ && c_1 = 1, && c_2 = \displaystyle \frac{1}{\theta}, \nonumber \\ & && \displaystyle c_3 = -\frac{\theta+1+\ln \theta}{\theta}, && \displaystyle c_4 = \frac{\theta+(\theta-1)\alpha -1}{2(\theta+1)}, \nonumber \\ & && \displaystyle c_5 = \frac{\alpha}{2}, && \displaystyle c_6 = \frac{\theta-\alpha-1}{2\theta}, \nonumber \\ & && \displaystyle c_7 = -\frac{\theta-\alpha-1}{2\theta}\ln \theta, && \displaystyle c_8 = -\frac{1}{6} + \frac{\alpha}{4} \left(\frac{1}{\theta}-1\right) + \frac{\alpha^2}{2}\left(\frac{1}{4\theta}+1\right). \label{eq: ci_j3} \end{align} (3.12) The precise values of the constants |$c_4$|⁠, |$c_7$|⁠, and |$c_8$| are not important for the proof of our results, but will play a role in the evaluation of the coefficient |$c^{(j)}$| of the logarithmic term and in further subleading terms in the Fredholm determinant expansion in (1.23). We now define \begin{equation} \label{eq: def_Gzeta} G(\zeta)=G^{(j)}(\zeta) := F^{(j)}\left(i s^{\rho^{(j)}} \zeta + \tau^{(j)}\right) {\rm e}^{-i s^{\rho^{(j)}} \left(\ln (s) \zeta - h^{(j)}(\zeta)\right)} \end{equation} (3.13) with \begin{equation} \label{eq: def_hzeta} h(\zeta)=h^{(j)}(\zeta) := -c_1^{(j)} \zeta \ln(i \zeta) - c_2^{(j)} \zeta \ln(-i \zeta) - c_3^{(j)} \zeta. \end{equation} (3.14) As |$s\to +\infty$| and |$\zeta$| such that |$s^\rho\zeta\to\infty$|⁠, we have by (3.9), \begin{equation}\label{as G} \ln G (\zeta)= c_4 \ln s + c_5 \ln \left(i \zeta \right)+ c_6 \ln \left(-i\zeta\right)+ c_7 + \frac{c_8}{is^\rho\zeta} + \mathcal{O}\left(\frac{1}{s^{2\rho}\zeta^2}\right). \end{equation} (3.15) On the other hand, if |$s\to +\infty$| and |$\zeta\to 0$| in such a way that |$s^\rho\zeta$| is bounded and such that |$i s^\rho \zeta + \tau$| is away from the poles of |$F$| (see (1.15)–(1.17)), we have \begin{gather}\label{lnGbdd} \ln G(\zeta) = \mathcal{O}\left(1\right). \end{gather} (3.16) The jump matrices in the RH problem for |$T$| can now be rewritten in terms of |$G$| and |$h$|⁠. We have \begin{equation} J_T(\zeta) = \begin{cases}\begin{pmatrix}1&- G(\zeta)e^{-is^{\rho}h(\zeta)}\\0&1 \end{pmatrix}, &\text{if}~ \zeta \in \Sigma_1\cup\Sigma_2, \\ \begin{pmatrix}1&0 \\ G(\zeta)^{-1}e^{is^{\rho}h(\zeta)} &1 \end{pmatrix}, & \text{if}~ \zeta \in \Sigma_3\cup\Sigma_4,\\ \begin{pmatrix} 1 & - G(\zeta)e^{-is^{\rho} h(\zeta)} \\ G(\zeta)^{-1} e^{is^{\rho} h(\zeta)} & 0\end{pmatrix}, & \text{if}~ \zeta \in \Sigma_5,\end{cases} \end{equation} (3.17) with contours |$\left\{\Sigma_j\right\}_{j=1,\ldots,5}$| as before (see Fig. 3). 3.3 Third transformation |$T \mapsto S$| We now proceed with a third transformation of the RH problem, where we introduce a |$g$|-function |$g(\zeta) = g^{(j)}(\zeta)$| with specific properties. We define the modified matrix \begin{equation} \label{eq: transfTS} S(\zeta) := {\rm e}^{s^{\rho} \frac{\ell}{2} \sigma_3} T(\zeta) {\rm e}^{-s^{\rho} \cdot g(\zeta) \sigma_3} {\rm e}^{-s^{\rho} \frac{\ell}{2} \sigma_3}, \end{equation} (3.18) where |$\ell=\ell^{(j)} \in \mathbb{C}$| is a constant which is to be determined. We would like to construct a |$g$|-function |$g(\zeta)=g^{(j)}(\zeta)$| which satisfies the following properties. Properties for the |$g$|-function (a) |$g$| is analytic in |$\mathbb{C} \setminus \Sigma_5$|⁠, (b) there exists a constant |$\ell$| such that |$g$| satisfies the relation \begin{equation}\label{eq: jump g} g_+(\zeta) + g_-(\zeta) - i h(\zeta) + \ell=0,\qquad \mbox{for}~ \zeta\in\Sigma_5, \end{equation} (3.19) (c) |$g$| has the asymptotics \begin{equation} \label{eq: cond2_gfun} g(\zeta) = \frac{g_1}{\zeta}+\mathcal{O}\left(\frac{1}{\zeta^2}\right), \qquad \text{as}~ \zeta \to \infty, \end{equation} (3.20) for some constant |$g_1$|⁠. Given such a |$g$|-function, one verifies that |$S$| solves the RH problem below. RH problem for |$S$| (a) |$S$| is analytic in |$\mathbb{C} \setminus (\Sigma_1 \cup\ldots\cup\Sigma_5)$|⁠; (b) for |$\zeta\in \Sigma_1 \cup\ldots\cup\Sigma_5$|⁠, we have |$S_+(\zeta) = S_-(\zeta) J_S(\zeta)$| with \begin{equation} \label{eq: jump_S} J_S(\zeta) = \begin{cases}\begin{pmatrix}1&- G(\zeta) e^{s^{\rho}(2 g(\zeta) - i h(\zeta) + \ell)}\\0&1 \end{pmatrix}, & \text{if}~ \zeta \in \Sigma_1\cup\Sigma_2, \\ \begin{pmatrix}1&0 \\ G(\zeta)^{-1} e^{-s^{\rho}(2 g(\zeta) - i h(\zeta) + \ell)} &1 \end{pmatrix}, & \text{if}~ \zeta \in \Sigma_3\cup\Sigma_4, \\ \begin{pmatrix}e^{-s^{\rho}(g_+(\zeta) - g_-(\zeta))} & - G(\zeta) \\ G(\zeta)^{-1} & 0\end{pmatrix}, & \text{if}~ \zeta \in \Sigma_5;\end{cases} \end{equation} (3.21) (c) |$\displaystyle S(\zeta) = I + \frac{S_1(s)}{\zeta}+{}\mathcal{O}\left(\frac{1}{\zeta^2}\right)$| as |$\zeta \to \infty$|⁠, \begin{equation}\label{S1} \text{with}~ \left(S_1(s)\right)_{2,2}=\left(U_1(s)\right)_{2,2}+s^\rho g_1. \end{equation} (3.22) Construction of the |$g$|-function. Instead of constructing the |$g$|-function directly, it turns out to be convenient to inspect its second derivative, and to impose some appropriate constraints to it afterwards. From the properties of |$g$|⁠, it is clear that |$g''$| needs to satisfy the following. Properties for |$g''$| (a) |$g''$| is analytic in |$\mathbb{C} \setminus \Sigma_5$|⁠, (b) |$g''$| satisfies the relation \begin{equation}\label{eq: jump g2} g''_+(\zeta) + g''_-(\zeta) = -i\frac{c_1+c_2}{\zeta},\qquad\mbox{for $\zeta\in\Sigma_5$,} \end{equation} (3.23) (c) as |$\zeta\to\infty$|⁠, there is a constant |$g_1$| such that \begin{equation} \label{eq: cond2_gfunsecder} g''(\zeta) = \frac{2g_1}{\zeta^3}+\mathcal{O}\left(\zeta^{-4}\right). \end{equation} (3.24) Given |$\Sigma_5$| with endpoints |$b_1$| and |$b_2=-\overline{b_1}$| (see Fig. 3), there is a unique function satisfying these properties, and which is such that \begin{equation*} r(\zeta)g''(\zeta)=\mathcal{O}(1), \quad \text{as}~ \zeta\to b_1 ~\text{and}~ \zeta \to b_2, \end{equation*} where \begin{equation}\label{def r} r(\zeta):=\left[(\zeta-b_1)(\zeta-b_2)\right]^{\frac{1}{2}}, \end{equation} (3.25) and the branch cut is chosen such that |$r(\zeta)$| is analytic in |$\mathbb{C}\setminus\Sigma_5$| and |$r(\zeta)\sim\zeta$| as |$\zeta\to\infty$|⁠. The unique function |$g''$| satisfying these properties is given by \begin{equation}\label{def g2} g''(\zeta)=-i\frac{c_1+c_2}{2}\left(\frac{1}{\zeta}-\frac{1}{r(\zeta)}+\frac{i ~{\rm {Im}}~ b_1}{\zeta r(\zeta)}\right). \end{equation} (3.26) By the asymptotic condition (3.24) for |$g''$|⁠, we can define \begin{equation}\label{def g g1} g'(\zeta):=\int_{\infty}^\zeta g''(\xi){\rm d}\xi,\qquad g(\zeta):=\int_\infty^\zeta g'(\xi){\rm d} \xi, \end{equation} (3.27) where the integration contour does not cross |$\Sigma_5$|⁠. The values of |$g'(\zeta)$| and |$g(\zeta)$| do not depend on the choice of integration contour. For arbitrary choices of the end points |$b_1, b_2$|⁠, the function |$g$| defined in this way does not satisfy the required properties for |$g$|⁠. Indeed, for |$\zeta \in \Sigma_5$| with |${\rm{Re}}~ \zeta<0$|⁠, we have \begin{equation}\label{id g1} g_+'(\zeta)+g_-'(\zeta)=-i(c_1+c_2)\left(\int_{b_1}^\zeta \frac{{\rm d}\xi}{\xi} +\int_\infty^{b_1}\left(\frac{1}{\xi}-\frac{1}{r(\xi)}+\frac{i~{\rm {Im}}~ b_1}{\xi r(\xi)}\right){\rm d}\xi\right). \end{equation} (3.28) Here, the integration from |$b_1$| to |$\zeta$| can be taken along |$\Sigma_5$| and the integration from |$\infty$| to |$b_1$| along the horizontal half-line from |$b_1-\infty$| to |$b_1$|⁠. On the other hand, by (3.19), we need that \begin{equation}\label{id g2} g_+'(\zeta)+g_-'(\zeta)=-ic_1\log(i\zeta)-ic_2\log(-i\zeta)-i(c_1+c_2+c_3),\qquad \zeta\in\Sigma_5. \end{equation} (3.29) Combining (3.28) and (3.29), we obtain after a straightforward calculation the identity \begin{align} &-i(c_1+c_2)\left(\log\zeta -\log\frac{\left|{\rm{Re}}~ b_1\right|}{2}-\frac{i\pi}{2}\sin\phi\right) +i(c_1+c_2)\sin\phi\, {\rm arcsinh}\left[\tan\phi\right] \notag\\ &\quad=-ic_1\log(i\zeta)-ic_2\log(-i\zeta)-i(c_1+c_2+c_3), \end{align} (3.30) where we define |$\phi$| by \begin{equation}\label{def b1b2} b_2 = -\overline{b_1}=be^{i\phi},\qquad \phi\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right). \end{equation} (3.31) Equating the real and imaginary parts of this equation, we obtain the value of the endpoints |$b_1, b_2$| from the equations \begin{align} &\sin\phi =\frac{c_2-c_1}{c_2+c_1},\\ \end{align} (3.32) \begin{align} &{\rm{Re}}~ b_1=-{\rm{Re}}~ b_2=-2 \left(\frac{c_2}{c_1}\right)^{-\frac{c_2-c_1}{2(c_2+c_1)}} {\rm e}^{-\frac{c_1+c_2+c_3}{c_1+c_2}}. \label{eq: reb2} \end{align} (3.33) Here we used the identities |$\rm{arcsinh} [\tan(\phi)] = \ln \left(\tan\left(\frac{\phi}{2} + \frac{\pi}{4}\right)\right)$| and |$\tan\left(\frac{\phi}{2} + \frac{\pi}{4}\right) = \sec(\phi) + \tan(\phi)$| which are valid for |$\phi\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$|⁠. For |$j = 1, 2$| we have that |$c_1 + c_2 + c_3 = 0$| such that (3.33) simplifies to read \begin{equation} \label{eq: reb2_j12} {\rm{Re}}~ b_1=-{\rm{Re}}~ b_2 = -2 \left(r-q\right)^{-\frac{r-q-1}{2(r-q+1)}}, \end{equation} (3.34) where |$q=0$| for the case |$j=1$|⁠. For |$j = 3$| we see that |$c_1 + c_2 + c_3 = - \frac{\ln(\theta)}{\theta}$| and hence \begin{equation} \label{eq: reb2_j3} {\rm{Re}}~ b_1=-{\rm{Re}}~ b_2 = -2 \theta^{\frac{3-\theta}{2(1+\theta)}}. \end{equation} (3.35) If now |$c_1 + c_2 + c_3 = 0$| and moreover |$c_1=c_2$| (this is true in the special case where either |$\theta=1$| or |$r=1, q=0$|⁠; in these cases the limiting kernel |$\mathbb K^{(j)}$| is the Bessel kernel), we have |$b_1,b_2\in \mathbb{R}$| and |$b_1 =-b_2 = -2$|⁠. In the cases we focus on, that is for |$j=1$|⁠, |$j=2$|⁠, and for |$j=3$| with |$\theta\leq 1$|⁠, we have |$c_2>c_1$| and thus |$\phi \geq 0$|⁠. For |$j = 3$| with |$\theta > 1$| on the other hand, we have |$c_2 < c_1$| and thus |$\phi < 0$|⁠. We return to this matter in Remark 3.2. Finally, the constant |$\ell$| can now be defined as |$\ell := i h(b_1) - 2 g(b_1)$|⁠, that is \begin{equation} \label{def: ell constant} \ell = i h(b_1) - 2 \int_\infty^{b_1} g'(\xi) {\rm d} \xi. \end{equation} (3.36) We now have the following lemma. Lemma 3.1 Let |$b_1, b_2$| be given by (3.31) and suppose that |$\phi\geq 0$|⁠. Let |$\Sigma_1,\ldots, \Sigma_5$| be as in (3.5), see also Figure 3. Then the following inequalities hold: \begin{align} &{\rm{Re}}~ \left[g_+(\zeta) - g_-(\zeta)\right] > 0,&& \zeta \in \Sigma_5 \setminus \left\{b_1,b_2 \right\}, \label{eq: cond4_gfun} \\ \end{align} (3.37) \begin{align} & {\rm{Re}}~\left[2g(\zeta) - i h(\zeta) - \ell \right] < 0, && \zeta \in \Sigma_1\cup\Sigma_2, \label{eq: cond5_gfun} \\ \end{align} (3.38) \begin{align} & {\rm{Re}}~\left[2g(\zeta) - i h(\zeta) - \ell \right] > 0,&& \zeta \in \Sigma_3\cup\Sigma_4. \label{eq: cond6_gfun} \end{align} (3.39) □ Proof. For |$\zeta \in \Sigma_5$|⁠, define \begin{equation} \varphi(\zeta) := g_+ (\zeta) - g_-(\zeta). \end{equation} (3.40) Since \begin{gather} g''(\zeta) = -i\frac{c_1+c_2}{2} \left(\frac{1}{\zeta} - \frac{1}{r(\zeta)} + \frac{i ~{\rm {Im}}(b_1)}{\zeta \, r(\zeta)} \right) \quad \text{with}~ r(\zeta) = \left[(\zeta-b_1)(\zeta-b_2)\right]^{\frac{1}{2}}, \end{gather} (3.41) we have \begin{gather} \varphi''(\zeta) = g_+''(\zeta) - g_-''(\zeta) = i (c_1+c_2) \frac{\zeta - i ~{\rm {Im}}(b_1)}{\zeta \, r_+(\zeta)}. \end{gather} (3.42) By the symmetry of the RH problem and the |$g$|-function, it is sufficient to prove (3.37) for |$\zeta \in (b_1,0) \subseteq \Sigma_5$|⁠. First of all, we notice that \begin{gather} {\rm{Re}}~ \left[\varphi(\zeta)\right] = {\rm{Re}}~ \left[\int_{b_1}^\zeta \int_{b_1}^\xi \varphi''(\eta) \, {\rm d} \eta \, {\rm d} \xi\right]. \end{gather} (3.43) Therefore, in order to get (3.37), we only need to prove that |$\arg\left[\varphi''(\zeta)\, {\rm d} \eta\,{\rm d} \xi\right]$| belongs to the right half plane, \[\arg\left[\varphi''(\zeta)\, {\rm d}\eta\,{\rm d}\xi\right]=\arg \left[i (c_1+c_2) \frac{\zeta - i ~{\rm {Im}}(b_1)}{\zeta \, (\zeta-b_1)_+^{1/2}(\zeta-b_2)_+^{1/2}}\, {\rm d}\eta\,{\rm d}\xi\right]\in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right).\] This is easily achieved, since \begin{align*} &\arg\left[i(c_1+c_2)\right]=\frac{\pi}{2}, &&\arg\left[{\rm d}\eta\,{\rm d}\xi\right]=-2\phi, \\ &\arg \left[\frac{1}{\zeta}\right]=-\pi+\phi, && \arg\left[(\zeta-b_1)_+^{-1/2}\right]=\frac{\phi}{2},\\ &\arg\left[(\zeta-b_2)_+^{-1/2} \right] \in \left(-\frac{\pi+\phi}{2}, -\frac{\pi}{2} \right), &&\arg\left[\zeta - i~{\rm {Im}}(b_1)\right]\in \left(-\pi, -\frac{\pi}{2}\right), \end{align*} which implies that \[\arg\left[\varphi''(\zeta)\, {\rm d}\eta\,{\rm d}\xi\right]\in \left(-\phi, \frac{\pi}{2}-\frac{\phi}{2}\right), \quad \text{with}~ \phi \in \left[0, \frac{\pi}{2} \right).\] Next, we want to show that the quantity |${\rm{Re}}~ \left[2g(\zeta) -ih(\zeta) + \ell \right]$| is positive on |$\Sigma_3\cup\Sigma_4$| and negative on |$\Sigma_1\cup\Sigma_2$|⁠. We focus again only on the parts of the contours lying in the left half of the complex plane. By construction of the |$g$|-function, we have |$g_+(\zeta) + g_-(\zeta) -ih(\zeta) + \ell =0$| on |$\Sigma_5$|⁠, hence \begin{equation} 2g_+(\zeta) -ih(\zeta) + \ell = g_+(\zeta) - g_-(\zeta) = \varphi(\zeta) \qquad \text{on}~\Sigma_5. \end{equation} (3.44) This implies that |$2g -ih + \ell $| is the analytic continuation of the function |$\varphi$| to the positive side of the curve |$\Sigma_5$|⁠, that is the region above |$\Sigma_5$|⁠. The second derivative of |$2g - i h + \ell$| is given by \begin{equation} 2g''(\zeta) - i h''(\zeta) = i (c_1+c_2) \frac{\zeta - i ~{\rm {Im}}(b_1)}{\zeta \, \check{r}(\zeta)} \end{equation} (3.45) with |$\check{r}(\zeta) = \left[(b_1-\zeta)(b_2-\zeta) \right]^{\frac{1}{2}}$| such that |$\check{r}$| is analytic on |$\mathbb{C} \setminus \left\{(b_1, b_1 - i \infty) \cup (b_2, b_2 - i \infty) \right\}$| and |$\check{r}(\zeta) \in i\mathbb{R}_+$| on the horizontal segment |$(b_1,b_2)$|⁠. For |$\zeta \in\Sigma_1$|⁠, we have \begin{gather} {\rm{Re}}~ \left[\varphi(\zeta)\right] = {\rm{Re}}~ \left[\int_\zeta^{b_1} \int_\xi^{b_1}\left(2g''(\eta) - ih''(\eta) \right) \, {\rm d} \eta \, {\rm d} \xi\right], \end{gather} (3.46) and it suffices to show that |$\arg \left[\left(2g''(\eta) - ih''(\eta)\right) \, {\rm d} \eta \, {\rm d} \xi\right]$| lies in the left half plane, meaning $$\arg \left[\left(2g''(\eta) - ih''(\eta)\right) \, {\rm d} \eta \, {\rm d} \xi\right] = \arg \left[i (c_1+c_2) \frac{\zeta - i ~{\rm {Im}}(b_1)}{\zeta \, \check{r}(\zeta)} \, {\rm d} \eta \, {\rm d} \xi\right] \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right).$$ This follows from \begin{align*} &\arg \left[i(c_1+c_2)\right]=\frac{\pi}{2}, &&\arg\left[{\rm d}\eta\,{\rm d}\xi\right]=2\pi -2\phi-2\epsilon, \\ &\arg \left[\frac{1}{\zeta}\right] \in \left(-\pi+\phi,-\pi+\phi+\epsilon \right), && \arg\left[(\zeta-b_1)^{-1/2}\right]=-\frac{\pi-\phi-\epsilon}{2},\\ &\arg\left[(\zeta-b_2)^{-1/2}\right] \in \left(-\frac{\pi}{2}, -\frac{1}{2}\arg \left[\zeta - i~{\rm {Im}}(b_1)\right] \right), &&\arg\left[\zeta - i~{\rm {Im}}(b_1)\right] \in \left(\pi-\phi-\epsilon,\pi\right), \end{align*} which implies that the argument lies in \[\left(\frac{3\pi}{2}-\frac{3\phi}{2}-\frac{5\epsilon}{2}, \frac{3\pi}{2}-\frac{\phi}{2}-\frac{\epsilon}{2}\right)\subset\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)\] for |$0 \leq \phi < \pi/2$| and |$0< \epsilon<\pi/10$|⁠. Finally, in order to prove that |${\rm{Re}}~ \left[2g(\zeta) -ih(\zeta) + \ell \right] > 0$| on |$\Sigma_3$|⁠, we need to show that $$\arg \left[i (c_1+c_2) \frac{\zeta - i ~{\rm {Im}}(b_1)}{\zeta \, (\zeta-b_1)^{1/2}(\zeta-b_2)^{1/2}} \, {\rm d}\eta\,{\rm d}\xi\right] \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right),$$ and this follows from \begin{align*} &\arg \left[i(c_1+c_2)\right] =\frac{\pi}{2}, &&\arg\left[{\rm d}\eta\, {\rm d}\xi\right]= 2\pi + 2\epsilon, \\ &\arg \left[\frac{1}{\zeta}\right] \in \left(-\pi -\epsilon,-\pi+\phi\right), && \arg \left[(\zeta-b_1)^{-1/2} \right] = -\frac{\pi + \epsilon}{2},\\ &\arg\left[(\zeta-b_2)^{-1/2}\right] \in \left(-\frac{1}{2}\arg\left[\zeta - i ~{\rm {Im}}(b_1) \right], - \frac{\pi}{2} \right), &&\arg\left[\zeta - i~{\rm {Im}}(b_1)\right] \in \left(\pi, \pi + \epsilon\right), \end{align*} implying that the argument belongs to the interval \begin{equation*} \left(- \frac{\pi}{2}, -\frac{\pi}{2} + \phi + \frac{5}{2}\epsilon \right)\subset\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \end{equation*} for |$0 \leq \phi < \pi/2$| and |$0< \epsilon<\pi/10$|⁠. ■ As a consequence of the above lemma, we obtain that the jump matrices |$J_S$| for |$S$| converge exponentially fast, as |$s\to +\infty$|⁠, to the identity matrix on |$\Sigma_1, \Sigma_2, \Sigma_3, \Sigma_4$|⁠, and that the diagonal of |$J_S$| converges to |$0$| exponentially fast on |$\Sigma_5$|⁠. This convergence is however not uniformly valid near the endpoints |$b_1$| and |$b_2$|⁠. 3.4 The global parametrix We look for an approximation to |$S$| that is valid for large |$s$| away from the endpoints |$b_1, b_2$|⁠. To that end, we want to find a matrix-valued function |$P^{\infty}(\zeta)$| satisfying the following RH conditions. RH problem for |$P^{\infty}$| (a) |${P}^{\infty}$| is analytic in |$\mathbb{C} \setminus \Sigma_5$|⁠; (b) |$P^{\infty}_+(\zeta) = P^{\infty}_-(\zeta) J^{\infty}(\zeta)$| with \begin{equation} \label{eq: jump_pinfty} J^{\infty}(\zeta) = \begin{pmatrix}0 & - G(\zeta) \\ G(\zeta)^{-1} & 0\end{pmatrix},\qquad \zeta\in\Sigma_5; \end{equation} (3.47) (c) as |$\zeta\to\infty$|⁠, we have \begin{equation}{P}^{\infty}(\zeta) = I + \frac{P^\infty_1(s)}{\zeta} + \mathcal{O}\left(\frac{1}{\zeta^2}\right). \label{Pinftyinfty} \end{equation} (3.48) In order to construct the solution, we first solve a similar and simpler RH problem with constant jumps. RH problem for |$Q^\infty$| (a) |$Q^{\infty}$| is analytic in |$\mathbb{C} \setminus \Sigma_5$|⁠; (b) $$ Q^{\infty}_+(\zeta) = Q^{\infty}_-(\zeta) \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$$ for |$\zeta\in\Sigma_5$|⁠; (c) |$ Q^{\infty}(\zeta) = I + \mathcal{O}\left(\frac{1}{\zeta}\right)$| as |$\zeta \to \infty$|⁠. The solution to this RH problem is explicit and it is given by (see [14, Chapter 7] for a similar construction) \begin{equation} Q^{\infty}(\zeta) = \frac{1}{2} \begin{pmatrix}\gamma(\zeta)+\gamma(\zeta)^{-1} & \frac{1}{i} \left(\gamma(\zeta)-\gamma(\zeta)^{-1}\right) \\-\frac{1}{i} \left(\gamma(\zeta)-\gamma(\zeta)^{-1}\right) & \gamma(\zeta)+\gamma(\zeta)^{-1}\end{pmatrix}, \end{equation} (3.49) where \begin{equation} \gamma(\zeta)=\left(\frac{\zeta-b_1}{\zeta - b_2}\right)^{1/4} \end{equation} (3.50) which is defined and analytic on |$\mathbb{C} \setminus \Sigma_5$|⁠, with branch cut on |$\Sigma_5$|⁠. Construction of |$P^\infty$| Now, |$P^\infty$| can be written in the form \begin{equation} \label{eq: Pinfty} P^{\infty}(\zeta) :={\rm e}^{-p_0 \sigma_3} Q^{\infty}(\zeta) {\rm e}^{p(\zeta) \sigma_3} \end{equation} (3.51) with |$p(\zeta)=p^{(j)}(\zeta)$| and |$p_0=p_0^{(j)}$| suitably defined as \begin{align} p(\zeta) &= -\frac{r(\zeta)}{2\pi i} \int_{\Sigma_5} \frac{\ln G(\xi)}{r_+(\xi)} \frac{{\rm d} \xi}{\xi-\zeta} \label{eq: pfunction}\\ \end{align} (3.52) \begin{align} p(\zeta) &= p_0 + \frac{p_1}{\zeta} + \mathcal{O}\left(\frac{1}{\zeta^2}\right) \qquad \text{as}~ \zeta \rightarrow \infty , \end{align} (3.53) with |$r(\zeta)$| as in (3.25), such that |$p(\zeta)$| satisfies \begin{equation}\label{jump p} p_+(\zeta) + p_-(\zeta) = -\ln G(\zeta), \end{equation} (3.54) for |$\zeta\in\Sigma_5$|⁠. After expanding (3.52) at |$\zeta = \infty$|⁠, we identify \begin{align} p_0 &= \frac{1}{2\pi i} \int_{\Sigma_5} \frac{\ln G(\xi)}{r_+(\xi)} {\rm d} \xi, \\ \end{align} (3.55) \begin{align} p_1 &= - \frac{b_1+b_2}{4\pi i} \int_{\Sigma_5} \frac{\ln G(\xi)}{r_+(\xi)} {\rm d} \xi + \frac{1}{2\pi i} \int_{\Sigma_5} \frac{\xi \ln G(\xi)}{r_+(\xi)} {\rm d}\xi \nonumber \\ &= \frac{1}{2\pi i} \int_{\Sigma_5} \frac{(\xi-i~{\rm {Im}}(b_1))\ln G(\xi)}{r_+(\xi)} {\rm d}\xi. \label{def p1} \end{align} (3.56) As |$\zeta\to b_k$|⁠, the solution |$P^\infty$| behaves like \begin{equation*} P^{\infty}(\zeta) =\mathcal{O}\left((\zeta -b_k)^{-\frac{1}{4}}\right),\qquad k=1,2. \end{equation*} 3.5 The local parametrix at the end points |$\boldsymbol{b_1,b_2}$| Near the end points |$b_1, b_2$| the global parametrix |$P^{\infty}$| cannot be a good approximation for |$S$|⁠, since it blows up, while |$S$| remains bounded. Hence, we need to introduce local parametrices near these points. The local parametrix |$P$| will be defined in small neighbourhoods of |$b_1$| and |$b_2$|⁠, \begin{equation*} \mathbb D_{\delta}(b_1) = \{z \in \mathbb{C} : |z - b_1| < \delta\},\qquad \mathbb D_{\delta}(b_2) = \{z \in \mathbb{C} : |z - b_2| < \delta\}, \end{equation*} for some small but fixed |$\delta > 0$|⁠, independent of |$s$|⁠. We will focus on the parametrix |$P$| near the end point |$b_1$|⁠. By symmetry, the parametrix near |$b_2$| will be given by |$P(\zeta)=\overline{P(-\overline{\zeta})}$|⁠. We will construct |$P$| in such a way that it has the same jumps as |$S$| in |$\mathbb D_{\delta}(b_1)$| and such that it matches with the global parametrix |$P^{\infty}$| on the circle |$\partial \mathbb D_{\delta}(b_1)$|⁠. The RH problem that we require |$P$| to satisfy is the following (see also Figure 4): Fig. 4. View largeDownload slide The jump contours for the local parametrix around the end point |$b_1$|⁠. Fig. 4. View largeDownload slide The jump contours for the local parametrix around the end point |$b_1$|⁠. RH problem for |$\boldsymbol{P}$| (a) |$P$| is analytic in |$\mathbb D_{\delta}(b_1) \setminus \left(\Sigma_5 \cup \Sigma_1\cup\Sigma_3\right)$|⁠; (b) $$P_+(\zeta) = P_-(\zeta) \begin{cases}\begin{pmatrix}1 & - G(\zeta) {\rm e}^{s^{\rho} \left(2g(\zeta) - ih(\zeta) +\ell\right)} \\ 0 & 1\end{pmatrix}, & z \in \Sigma_1 \cap \mathbb D_{\delta}(b_1) ,\\ \begin{pmatrix}1 & 0 \\ G(\zeta)^{-1} {\rm e}^{-s^{\rho} \left(2g(\zeta)- ih(\zeta) + \ell\right)} & 1\end{pmatrix}, & z \in \Sigma_3 \cap \mathbb D_{\delta}(b_1), \\ \begin{pmatrix}{\rm e}^{-s^{\rho} \left(g_+(\zeta) - g_-(\zeta) \right)} & - G(\zeta) \\ G(\zeta)^{-1} & 0\end{pmatrix}, & z \in \Sigma_5 \cap \mathbb D_{\delta}(b_1);\end{cases}$$ (c) |$\displaystyle P(\zeta) = P^{\infty}(\zeta)\left(I + o(1)\right)$| as |$s\to + \infty$| for |$\zeta \in \partial \mathbb D_{\delta}(b_1)$|⁠. In the next paragraph, we construct |$P$| explicitly in terms of the Airy function. The Airy model RH problem. We need a slight variation of the standard model RH problem associated to the Airy function which was used for instance in [14, 19, 20, 24]. Therefore, we follow [12, Section 3.5.1] and define \[ y_k(\zeta)={\rm e}^{\frac{2\pi i k}{3}}{\rm {Ai}}({\rm e}^{\frac{2\pi i k}{3}}\zeta),\qquad k=0,1,2, \] where |${\rm {Ai}}$| is the Airy function. Let |$A_1$|⁠, |$A_2$|⁠, |$A_3$| be entire functions given by \begin{align} & A_1(\zeta)=-i\sqrt{2\pi} \begin{pmatrix} -y_2(\zeta) & -y_0(\zeta)\\ -y_2'(\zeta) & -y_0'(\zeta) \end{pmatrix},\\ \end{align} (3.57) \begin{align} & A_2(\zeta)=-i\sqrt{2\pi} \begin{pmatrix} -y_2(\zeta) & y_1(\zeta)\\ -y_2'(\zeta) & y_1'(\zeta) \end{pmatrix}, \\ \end{align} (3.58) \begin{align} & A_3(\zeta)=-i\sqrt{2\pi} \begin{pmatrix} y_0(\zeta) & y_1(\zeta)\\ y_0'(\zeta) & y_1'(\zeta) \end{pmatrix}. \end{align} (3.59) Using the well-known Airy function identity |$y_0+y_1+y_2=0$|⁠, one verifies the relations \begin{align} \label{RHP A: b1} & A_1(\zeta)=A_2(\zeta) \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}, \\ \end{align} (3.60) \begin{align} & A_2(\zeta)=A_3(\zeta) \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, \\ \label{RHP A: b3} \end{align} (3.61) \begin{align} & A_1(\zeta)=A_3(\zeta) \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}. \end{align} (3.62) Moreover, by the asymptotic behaviour for the Airy function in the complex plane, we have that \begin{align}\label{RHP:A-c} A_k(\zeta) &= \zeta^{-\frac{\sigma_3}{4}}\begin{pmatrix}1&i\\1&-i\end{pmatrix} \left[I+O\left(\zeta^{-3/2}\right)\right] {\rm e}^{-\frac{2}{3}\zeta^{3/2}\sigma_3} \end{align} (3.63) as |$\zeta\to\infty$| in the sector |$S_k$| for |$k=1, 2, 3,$| with \begin{equation}\label{def sector Sk} S_k=\left\{\zeta\in\mathbb{C}: \frac{2k-3}{3}\pi+\delta\leq\arg\zeta\leq\frac{2k+1}{3}\pi-\delta \right\}, \qquad k=1, 2, 3, \end{equation} (3.64) for any |$\delta>0$|⁠. Construction of |$P$|⁠. We define the local parametrix in the following form, \begin{equation}\label{def P} P(\zeta) = E(\zeta) A_k\left(s^{\frac{2}{3} \rho}f(\zeta)\right) {\rm e}^{s^{\rho} q(\zeta) \sigma_3} G(\zeta)^{-\frac{\sigma_3}{2}}, \end{equation} (3.65) for |$\zeta$| in region |$[k]$| as shown in Figure 4: region |$[1]$| is the region between |$\Sigma_5$| and |$\Sigma_1$|⁠; region |$[2]$| is the one between |$\Sigma_1$| and |$\Sigma_3$|⁠; region |$[3]$| is the one between |$\Sigma_3$| and |$\Sigma_5$|⁠. Here, |$E$| is an analytic function in |$\mathbb D_{\delta}(b_1)$| which will be constructed below, |$q(\zeta)$| is an analytic function on |$\mathbb D_{\delta}(b_1) \setminus \Sigma_5$| given by \begin{equation} q(\zeta) := g(\zeta) - \frac{i}{2} h(\zeta) + \frac{\ell}{2}, \end{equation} (3.66) and |$f(\zeta)$| will be a conformal map from |$\mathbb D_{\delta}(b_1)$| to a neighbourhood of |$0$| which we will determine below. In order to achieve the RH conditions (a)–(c), we need to define the analytic prefactor |$E$| and the conformal map |$f$| appropriately. The conformal map and the analytic prefactor. First, we need that |$f$| is such that it maps region |$[k]$|⁠, for |$k=1,2,3$|⁠, to a subset of the region |$S_k$| defined in (3.64). If this is true, we can use the asymptotic behaviour (3.63) of the functions |$A_k$| to conclude from (3.65) that \begin{align}\label{matching P1} P(\zeta)=E(z)\left(s^{\frac{2}{3}\rho}f(\zeta)\right)^{-\frac{\sigma_3}{4}} \begin{pmatrix}1&i\\1&-i\end{pmatrix} \left[I+O\left(s^{-\rho}\right)\right] \times {\rm e}^{-\frac{2s^\rho}{3}f(\zeta)^{3/2}\sigma_3}{\rm e}^{s^{\rho} q(\zeta) \sigma_3} G(\zeta)^{-\frac{\sigma_3}{2}} \end{align} (3.67) for |$\zeta\in\partial \mathbb D_{\delta}(b_1)$| as |$s\to + \infty$|⁠, if |$\delta>0$| is sufficiently small. This has to be |$P^{\infty}(\zeta)(1+o(1))$|⁠, which suggests us to take |$f$| and |$E$| as follows, \begin{align} &f(\zeta)=\left(\frac{3}{2}q(\zeta)\right)^{2/3},\\ \end{align} (3.68) \begin{align} &E(\zeta)= P^\infty(\zeta) G(\zeta)^{\frac{\sigma_3}{2}} \begin{pmatrix}1&i\\1&-i\end{pmatrix}^{-1} \left(s^{\frac{2}{3} \rho}f(\zeta)\right)^{\frac{\sigma_3}{4}}. \end{align} (3.69) First, it can be verified using the jump relation for |$P^\infty$| and by taking into account the branch cuts of the roots that |$E$| is indeed analytic in |$\mathbb D_{\delta}(b_1)$|⁠. Secondly, using the properties of the |$g$|-function, namely (3.26), we have as |$z \to b_1$|⁠, \begin{equation} q''(z) = - \frac{(c_1 + c_2)}{2 \sqrt{2}} \frac{\sqrt{\left|{\rm{Re}}(b_1)\right|}}{b_1} (z - b_1)^{-\frac{1}{2}} + \mathcal{O}\left((z - b_1)^{\frac{1}{2}}\right) \end{equation} (3.70) and hence \begin{equation} q(z) = -\frac{2}{3} \frac{(c_1 + c_2)}{\sqrt{2}} \frac{\sqrt{\left|{\rm{Re}}(b_1)\right|}}{b_1} (z - b_1)^{\frac{3}{2}} + \mathcal{O}\left((z - b_1)^{\frac{5}{2}}\right). \end{equation} (3.71) It follows that |$f$| is indeed a conformal map and |$f'(b_1) \in \mathbb{C}$| with \begin{equation} \label{eq: der_conformalmap_b1} \text{arg}\left[f'(b_1)\right]=\frac{2\phi}{3} \in \left[0, \frac{\pi}{3}\right), \ \text{since}~ \phi \in \left[0, \frac{\pi}{2}\right). \end{equation} (3.72) We can now verify that |$f$| maps the regions |$[1]$|–|$[3]$| from Figure 4 to the admissible sectors |$S_1, S_2, S_3$| in (3.64). From (3.72), it follows indeed that region |$[1]$|⁠, where |$-\phi<\arg\left[\zeta-b_1 \right] <\pi-\phi-\epsilon$|⁠, is mapped into the sector |$S_1$|⁠, that region |$[2]$|⁠, defined by |$\pi-\phi-\epsilon<\arg\left[\zeta-b_1\right] <\pi+\epsilon$|⁠, is mapped into the sector |$S_2$|⁠, and that region |$[3]$|⁠, where |$-\pi+\epsilon<\arg\left[\zeta-b_1\right]<-\phi$|⁠, is mapped into the sector |$S_3$|⁠. The jump relation (b) for |$P$| is now easily verified. 3.6 Final transformation |$\boldsymbol{S \mapsto R}$| For the final transformation we define \begin{equation} \label{eq: transfSR} R(\zeta) = S(\zeta) \begin{cases}\left(P(\zeta)\right)^{-1} & \text{if}~ \zeta \in \mathbb{D}_{\delta}(b_1) \cup \mathbb{D}_{\delta}(b_2) \\ \left(P^{\infty}(\zeta)\right)^{-1} & \text{elsewhere}\end{cases}. \end{equation} (3.73) It follows that |$R(\zeta)$| satisfies the following RH problem: RH problem for |$R$| (a) |$R$| is analytic in |$\mathbb{C} \setminus \Gamma_R$| (see Fig. 5 for the definition of the contour |$\Gamma_R$|⁠); (b) |$R_+(\zeta) = R_-(\zeta) J_R(\zeta)$| for |$\zeta \in \Gamma_R$| with \begin{equation} J_R(\zeta) = \begin{cases}P_-^{\infty}(\zeta) J_S(\zeta) \left(P_+^{\infty}(\zeta)\right)^{-1} & \text{if}~ \zeta \in \Gamma_R \setminus (\partial\mathbb{D}_{\delta}(b_1) \cup \partial \mathbb{D}_{\delta}(b_2)) \\ P(\zeta) \left(P^\infty(\zeta)\right)^{-1} & \text{if}~ \zeta \in \partial \mathbb{D}_{\delta}(b_1) \cup \partial \mathbb{D}_{\delta}(b_2),\end{cases} \end{equation} (3.74) where |$J_S(\zeta)$| is given by (3.21), and where we choose the clockwise orientation for the circles around |$b_1, b_2$|⁠; (c) as |$\zeta \to \infty$| \begin{equation} \displaystyle R(\zeta) = I + \frac{R_1(s)}{\zeta} + \mathcal{O}\left(\frac{1}{\zeta^2}\right). \label{Rinfty} \end{equation} (3.75) Fig. 5. View largeDownload slide The contour |$\Gamma_R$| for the RH problem for |$R(\zeta)$|⁠. Fig. 5. View largeDownload slide The contour |$\Gamma_R$| for the RH problem for |$R(\zeta)$|⁠. By construction, the jump matrix for |$R$| is close to the identity matrix as |$s \to + \infty$| uniformly in |$\zeta$|⁠. We have \begin{equation} J_R(\zeta) = \begin{cases}I + \frac{J_R^{(1)}(\zeta)}{s^\rho}+ \mathcal{O}\left(s^{-2\rho}\right) & \text{if}~ \zeta \in \partial \mathbb{D}_{\delta}(b_1) \cup \partial \mathbb{D}_{\delta}(b_2), \\ I + \mathcal{O}\left(\frac{e^{- c s^{\rho}}}{|\zeta| + 1}\right) & \text{elsewhere,}\end{cases} \label{jumpforR} \end{equation} (3.76) for some fixed constant |$c > 0$| and for some function |$J_R^{(1)}(\zeta)$| independent of |$s$|⁠, uniformly in |$\zeta$|⁠. Hence, by standard arguments for small norm RH problems (see e.g., [28, Section 5.1.3]), it follows that \begin{equation}\label{expansion R s} R(\zeta) = I + \frac{R^{(1)}(\zeta)}{s^\rho}+\mathcal{O}\left(s^{-2\rho}\right) \qquad \text{as}~ s \to + \infty, \end{equation} (3.77) uniformly for |$\zeta\in \mathbb{C}\setminus\Gamma_R$|⁠. The error term |$R^{(1)}(\zeta)$| can be computed explicitly in terms of |$J_R^{(1)}$| (see [21]), and as |$\zeta\to\infty$| it behaves like \begin{equation}\label{expansion R1 zeta}R^{(1)}(\zeta)=\frac{R_1^{(1)}}{\zeta}+\mathcal{O}\left(\frac{1}{\zeta^{2}}\right), \end{equation} (3.78) for some constant matrix |$R_1^{(1)}$|⁠. In conclusion, the asymptotic value of |$R_1(s)$| defined in (3.75) is equal to \begin{equation}\label{expansion R1 s} R_1(s) = \frac{R_1^{(1)}}{s^\rho}+\mathcal{O}\left(s^{-2\rho}\right) \qquad \text{as}~ s \to + \infty. \end{equation} (3.79) Remark 3.2. In the RH analysis, we restricted ourselves to the cases |$j=1, 2$| and |$j=3$| with |$\theta\leq 1$|⁠. If |$j=3$| and |$\theta>1$|⁠, some modifications are required. Since the angle |$\phi$| in (3.31) becomes negative, the endpoints |$b_1, b_2$| lie in the lower half plane and the jump contours in the RH problems for |$T$| and |$S$| need to be changed. Compared to the contours in Figure 3, all contours have to be mirrored with respect to the real line, in particular we need to take \[\Sigma_5\mapsto \overline{\Sigma_5},\ \Sigma_1\mapsto\overline{\Sigma_3},\ \Sigma_2\mapsto\overline{\Sigma_4},\ \Sigma_3\mapsto\overline{\Sigma_1},\ \Sigma_4\mapsto\overline{\Sigma_2}.\] With these modified contours, it is straightforward to adapt the proof of Lemma 3.1 to the case where |$\phi<0$|⁠. The rest of the analysis, namely the construction of the global and local parametrices and of the small norm RH problem for |$R$|⁠, is similar to the case |$\phi\geq 0$|⁠. □ 4 The Fredholm Determinant We can now invert all the transformations |$Y \mapsto U \mapsto T \mapsto S \mapsto R$| and find an explicit asymptotic expression for the Fredholm determinant as |$s\to +\infty$|⁠. From (3.3), we know that \begin{equation*} \frac{{\rm d}}{{\rm d} s} \ln \det\left(1 - \mathbb{K}^{(j)}\bigg|_{[0, s]}\right) = - i s^{\rho^{(j)} - 1} \left(U_1(s)\right)_{2, 2}. \end{equation*} Combining (3.22), (3.48), (3.51), and (3.75), we obtain \begin{align} \frac{{\rm d}}{{\rm d} s} \ln \det\left(1 - \mathbb{K}^{(j)}\bigg|_{[0, s]}\right) &= - i s^{\rho - 1} \left(\left(S_1(s)\right)_{2,2} - s^{\rho} g_1\right) \nonumber \\ &= i g_1 s^{2 \rho - 1} - i s^{\rho - 1} \left(\left(P^{\infty}_1(s)\right)_{2,2} + \left(R_1(s) \right)_{2,2} \right) \nonumber \\ &= i g_1 s^{2 \rho - 1} - i s^{\rho - 1} \left(-p_1(s) + \left(R_1(s) \right)_{2,2} \right). \label{Fredholm expansion 1} \end{align} (4.1) The large |$s$| asymptotics for |$\left(R_1(s) \right)_{2,2}$| are given in (3.79), and we will now compute |$g_1(s)$| and |$p_1(s)$|⁠. 4.1 Calculation of |$\boldsymbol{g_1}$| We recall the definition of the second derivative of the |$g$|-function (see (3.26)), \begin{equation}\label{def g2 bis} g''(\zeta)=-i\frac{c_1+c_2}{2}\left(\frac{1}{\zeta}-\frac{1}{r(\zeta)}+\frac{i ~{\rm {Im}}~ b_1}{\zeta r(\zeta)}\right), \end{equation} (4.2) where as before \begin{equation} r(\zeta):=\left[(\zeta-b_1)(\zeta-b_2)\right]^{\frac{1}{2}}. \end{equation} (4.3) In order to calculate |$g_1$|⁠, we expand (4.2) as |$\zeta \to \infty$| and obtain \begin{align} g''(\zeta) &= -\frac{i(c_1+c_2)}{4\zeta^3} \left[b_1b_2 + \left(~{\rm {Im}}(b_1)\right)^2 \right] + \mathcal{O}\left(\frac{1}{\zeta^4}\right) \nonumber \\ &= \frac{i (c_1+c_2) \left({\rm{Re}}(b_1)\right)^2}{4 \zeta^3} + \mathcal{O}\left(\frac{1}{\zeta^4}\right). \end{align} (4.4) On the other hand, |$g(\zeta) := \frac{g_1}{\zeta} + \mathcal{O}\left(\zeta^{-2}\right)$| as |$\zeta \rightarrow \infty$| (see (3.20)), which implies \begin{equation} g''(\zeta) = \frac{2g_1}{\zeta^3} + \mathcal{O}\left(\frac{1}{\zeta^4}\right). \end{equation} (4.5) Therefore, we can identify the coefficient |$g_1$| as \begin{equation} g_1 = \frac{i \left({\rm{Re}}(b_1)\right)^2 (c_1+c_2)}{8}. \label{g1explicit} \end{equation} (4.6) 4.2 Calculation of |$\boldsymbol{p_1(s)}$| We recall from (3.56) that \begin{equation} p_1 (s)= \frac{1}{2\pi i} \int_{\Sigma_5} \frac{(\xi-i~{\rm {Im}}(b_1)) \ln G(\xi)}{r_+(\xi)} {\rm d}\xi. \end{equation} (4.7) We can split the integration over |$\Sigma_5 = [b_1,0] \cup [0,b_2]$| into integration over |$C_1$| and |$C_2$| with \[ C_1 = \Sigma_5 \cap \left\{\left|\zeta \right| < M s^{-\rho} \right\},\qquad C_2= \Sigma_5 \setminus C_1, \] for some fixed sufficiently large |$M>0$|⁠. By (3.16), one checks that the contribution of the integrals over |$C_1$| to |$p_1$| can be written as |$\frac{a_1(M)}{s^\rho}+\mathcal{O}\left(s^{-2\rho}\right)$| as |$s\to +\infty$|⁠, for some constant |$a_1(M)$| depending on |$M$|⁠. On the other hand, for the integral over |$C_2$| we can use (3.15), and obtain \begin{align} p_1 (s)= \int_{C_2} \frac{(\xi-i~{\rm {Im}}(b_1)) \left[c_4 \ln s + c_5 \ln \left(i \xi \right)+ c_6 \ln \left(-i\xi\right)+ c_7 \right]}{r_+(\xi)} \frac{{\rm d}\xi}{2\pi i} \\ \qquad + s^{-\rho} \left[- \frac{c_8}{2\pi} \int_{C_2} \frac{(\xi - i~{\rm {Im}}(b_1)){\rm d} \xi}{\xi\, r_+(\xi)} +a_1(M)\right] + \mathcal{O}\left(\frac{1}{s^{2\rho}}\right) \end{align} (4.8) as |$s \to + \infty$|⁠. We now replace the first two integrals over |$C_2$| again by integrals over the whole contour |$\Sigma_5$|⁠; this implies adding a contribution of order |$s^{-\rho}$| (possibly depending on |$M$|⁠) which will be counted in the |$s^{-\rho}$| term in the formula below. We get \begin{align} p_1(s) &= (c_4\ln s + c_7) \int_{\Sigma_5} \frac{\xi -i~{\rm {Im}}(b_1)}{r_+(\xi)} \frac{{\rm d}\xi}{2\pi i} \nonumber \\ & \qquad+ c_5 \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1)) \ln \left(i \xi \right)}{r_+(\xi)} \frac{{\rm d}\xi}{2\pi i} + c_6 \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1)) \ln \left(-i\xi\right)}{r_+(\xi)} \frac{{\rm d}\xi}{2\pi i} \nonumber \\ & \qquad + s^{-\rho} \left[- \frac{c_8}{2\pi} \int_{C_2} \frac{(\xi - i~{\rm {Im}}(b_1)){\rm d} \xi}{\xi\, r_+(\xi)} +a_1(M) + a_2(M)\right] + \mathcal{O}\left(\frac{1}{s^{2\rho}}\right) \nonumber \\ & =: \left(c_4 \ln s + c_7\right) I_1 + c_5 I_2 + c_6 I_3 + \frac{\mathcal{K}}{s^{\rho}} + \mathcal{O}\left(\frac{1}{s^{2\rho}}\right),\label{p1Ij} \end{align} (4.9) for some constant |$\mathcal K$|⁠. The value of |$\mathcal K$| depends on the parameters |$\{\nu_j \}$|⁠, |$\{\mu_k\}$|⁠, |$\alpha$|⁠, |$\theta$| but we do not compute its explicit value. Note that |$\mathcal K$| does not depend on |$M$|⁠, although it may seem to a priori, since |$p_1(s)$| does not depend on |$M$|⁠. The integrals |$I_1,I_2, I_3$| are defined as \begin{align*} &I_1 = \int_{\Sigma_5} \frac{\xi -i~{\rm {Im}}(b_1)}{r_+(\xi)} \frac{{\rm d}\xi}{2\pi i},\\ &I_2 = \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1)) \ln \left(i \xi \right)}{r_+(\xi)} \frac{{\rm d} \xi}{2 \pi i},&& I_3 = \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1)) \ln \left(-i \xi \right)}{r_+(\xi)} \frac{{\rm d} \xi}{2 \pi i}, \end{align*} and they remain to be computed. Computation of |$\boldsymbol{I_1}$|⁠. We assume that the end points |$b_1, b_2$| lie in the upper half-plane, as set in Section 3.1 (for the other case, the argument is similar). By analyticity we can deform the contour |$\Sigma_5$| to a horizontal segment between |$b_1$| and |$b_2$| and we can easily show that |$I_1$| is zero: \begin{equation} I_1 = \int_{\Sigma_5} \frac{(\xi - i~{\rm {Im}}(b_1)){\rm d}\xi}{2\pi ir_+(\xi)}=-\int_{{\rm{Re}}~ (b_1)}^{{\rm{Re}}~ (b_2)} \frac{u}{\sqrt{{\rm{Re}}(b_1)^2-u^2}} \frac{{\rm d} u}{2\pi}=0 \label{I1} \end{equation} (4.10) by symmetry. Computation of |$I_3$|⁠. For the integral |$I_3$|⁠, we use again analyticity to deform as before the contour |$\Sigma_5$| into the segment |$[b_1, b_2]$| (the logarithmic branch cut lies on |$i\mathbb{R}^-$|⁠): we obtain \begin{align} I_3 = \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1))\ln \left(-i \xi \right)}{r_+(\xi)} \frac{{\rm d} \xi}{2\pi i} = -\int_{{\rm{Re}}(b_1)}^{{\rm{Re}}(b_2)} \frac{u\ln \left(-iu +~{\rm {Im}}(b_1) \right)}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 - u^2}} \frac{{\rm d} u}{2\pi} \\ = -\frac{i}{\pi} \int_{0}^{{\rm{Re}}(b_2)} \frac{u\arg\left[-iu + ~{\rm {Im}}(b_1) \right]}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 - u^2}} {\rm d} u . \end{align} (4.11) The last integral is equal to |$-\frac{\pi}{2}(|b_1|- {\rm {Im}}(b_1))$|⁠. Therefore, \begin{equation} I_3 = \frac{i}{2}(|b_1|-~ {\rm {Im}}(b_1)). \label{I3} \end{equation} (4.12) Computation of |$\boldsymbol{I_2}$|⁠. For the integral |$I_2$|⁠, the branch cut of the logarithm is on |$i\mathbb{R}^+$|⁠, and the integration over |$\Sigma_5$| can be deformed by analyticity to a contour as shown in Fig. 6. Fig. 6. View largeDownload slide Deformation of the contour for the integral |$I_2$|⁠. Fig. 6. View largeDownload slide Deformation of the contour for the integral |$I_2$|⁠. Given a parametrization of the form |$\xi = i ~{\rm {Im}}(b_1) + u$|⁠, |$u \in [{\rm{Re}}(b_1), {\rm{Re}}(b_2)]$| for the horizontal parts and the parametrization |$\xi = iv$|⁠, |$v \in [0, ~{\rm {Im}}(b_1)] $| for the vertical parts, we have \begin{align} I_2 &= \int_{\Sigma_5} \frac{(\xi -i~{\rm {Im}}(b_1))\ln \left(i \xi \right)}{r_+(\xi)} \frac{{\rm d} \xi}{2\pi i} \nonumber \\ & = -\int_{{\rm{Re}}(b_1)}^{{\rm{Re}}(b_2)} \frac{u\ln \left(iu - {\rm {Im}}(b_1) \right)}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 - u^2}} \frac{{\rm d} u}{2\pi} + \int_0^{~{\rm {Im}}(b_1)} \frac{(v - ~{\rm {Im}}(b_1))(\ln_+(-v) - \ln_-(-v))}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 + \left(v- {\rm {Im}}(b_1)\right)^2}}\frac{{\rm d} v}{2\pi} \nonumber \\ & = -\frac{i}{\pi} \int_{0}^{{\rm{Re}}(b_2)} \frac{u\arg\left[iu - ~{\rm {Im}}(b_1) \right]}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 - u^2}} {\rm d} u+ i \int_0^{~{\rm {Im}}(b_1)} \frac{v- {\rm {Im}}(b_1)}{\sqrt{\left({\rm{Re}}(b_1)\right)^2 + \left(v- {\rm {Im}}(b_1)\right)^2}}{\rm d} v. \end{align} (4.13) The second integral is equal to |$-|b_1|-{\rm{Re}}(b_1)$|⁠. For the first integral, we note that \[\arg\left[iu - ~{\rm {Im}}(b_1) \right]=\arg\left[-iu + ~{\rm {Im}}(b_1) \right]+{\rm {sgn}} (u) \pi,\] and then it follows that \begin{equation}\label{I2 final} I_2=I_3-i|b_1|= \frac{i}{2}(|b_1|- {\rm {Im}}(b_1))-i|b_1|. \end{equation} (4.14) In conclusion, substituting (4.10), (4.14) and (4.12) (4.9), we get as |$s \to + \infty$|⁠, \begin{equation} p_1(s) =- i c_5 |b_1| +i \frac{c_5+c_6}{2} \left(|b_1|- {\rm {Im}}(b_1) \right) + \frac{\mathcal{K}}{s^{\rho}} + \mathcal{O}\left(\frac{1}{s^{2\rho}}\right) \label{p1explicit} \end{equation} (4.15) as |$s \to + \infty$|⁠. 4.3 The final asymptotic expansion of the Fredholm determinant Using (3.79), (4.6) and (4.15) in (4.1), we obtain \begin{align*} \frac{{\rm d}}{{\rm d} s} \ln \det\left(1 - \mathbb{K}^{(j)}\bigg|_{[0, s]}\right)& = - \frac{\left({\rm{Re}}(b_1)\right)^2 (c_1+c_2)}{8} s^{2 \rho - 1} \\ &\quad - \left(-c_5 |b_1| + \frac{c_5+c_6}{2} \left(|b_1|- {\rm {Im}}(b_1) \right) \right) s^{\rho - 1} \nonumber \\ &\quad+ \frac{-\mathcal{K} + \left(R_1^{(1)}\right)_{2,2}}{is} + \mathcal{O}\left(s^{-\rho-1}\right),\qquad s\to + \infty . \end{align*} Integrating in |$s$|⁠, we obtain \begin{equation} \ln \det\left(1 - \mathbb{K}^{(j)}\bigg|_{[0, s]}\right) = - a^{(j)} \, s^{2 \rho} +b^{(j)} \, s^{\rho} + c^{(j)}\, \ln s + \ln C^{(j)} +\mathcal{O}\left(s^{-\rho}\right),\qquad s\to + \infty, \end{equation} (4.16) for some integration constant |$\ln C^{(j)}$|⁠, and with \begin{align} &a^{(j)}=\frac{\left({\rm{Re}}(b_1)\right)^2 (c_1+c_2)}{16\rho},\\ \end{align} (4.17) \begin{align} &b^{(j)}=-\frac{1}{\rho} \left(- c_5 |b_1| + \frac{c_5+c_6}{2} \left(|b_1|- {\rm {Im}}(b_1) \right) \right). \end{align} (4.18) Combining (3.34) and (3.35) with the specific values for the constants |$\{c_i\}$| from (3.10), (3.11) and (3.12) we immediately get Theorem 1.2. Remark 4.1. The values of |$\mathcal{K}$| and |$\left(R^{(1)}_{1}\right)_{2,2}$| will determine the coefficient |$c^{(j)}$| in front of the logarithmic term in the asymptotic expansion of the Fredholm determinant. As already stressed in the introduction, their value can in principle be explicitly computed, but the computations are quite involved, and we do not proceed with this. □ Funding This work was supported by European Research Council under the European Union’s Seventh Framework Programme (FP/2007/2013)/ ERC Grant Agreement 307074 to T.C. and M.G; FWO Flanders Project G.0934.13 and KU Leuven Research Grant OT/12/073 to D.S.; and Belgian Interuniversity Attraction Pole P07/18. Appendix. Limit of the Smallest Eigenvalue Distribution as a Fredholm Determinant Correlation kernels and scaling limits. We first express the finite |$n$| correlation kernels |$K_n^{(j)}$| as double contour integrals. 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Samuelson,, Peter

doi: 10.1093/imrn/rnx198pmid: N/A

Abstract We give a topological realization of the (spherical) double affine Hecke algebra |$\mathrm{SH}_{q,t}$| of type |${\mathfrak{sl}}_2$|⁠, and we use this to construct a module over |$\mathrm{SH}_{q,t}$| for any knot |$K \subset S^3$|⁠. As an application, we give a purely topological interpretation of Cherednik’s two-variable polynomials |$P_n(r,s; q,t)$| of type |${\mathfrak{sl}}_2$| from [14] (where |$r,s \in {\mathbb{Z}}$| are relatively prime). We then generalize the construction of these polynomials (for |${\mathfrak{sl}}_2$|⁠) from torus knots to all iterated cables of the unknot and prove they specialize to the colored Jones polynomials of the knot. Finally, in the Appendix we compare this construction to a later construction of Cherednik and Danilenko. 1 Introduction The (spherical) double affine Hecke algebra|$\mathrm{SH}_{q,t}({\mathfrak{g}})$| is a (noncommutative) algebra associated to a semisimple complex Lie algebra |${\mathfrak{g}}$| and two parameters |$q,t \in {\mathbb{C}}^*$|⁠. These algebras were introduced by Cherednik to prove Macdonald’s conjectures about Macdonald polynomials [11], and they have since found applications in many areas of mathematics (see, e.g., [12] and references therein). In this article, we give two new connections between double affine Hecke algebras (DAHAs) and knots—the first involves iterated cables of the unknot, and the second involves arbitrary knots. 1.1 Cables In [3] (see also [2]), Aganagic and Shakirov used refined Chern–Simons theory and generalized Verlinde algebras to construct |$q,t$| versions of Reshetikhin–Turaev invariants for torus knots. Cherednik [14] gave a construction of these polynomials using double affine Hecke algebras. More precisely, he used the representation theory of |$\mathrm{SH}_{q,t}({\mathfrak{g}})$| to construct polynomials |$P_{\alpha,r,s}(q,t) \in {\mathbb{C}}[q^{\pm 1},t^{\pm 1}] $|⁠, where |$\alpha \in \mathfrak t^*$| is an integral dominant weight and |$r,s \in {\mathbb{Z}}$| are relatively prime. He gave a number of conjectural properties of these polynomials—the conjecture which is relevant to us is that they specialize at |$t=q$| to the Reshetikhin–Turaev invariants of the |$(r,s)$| torus knot. From now on we fix |${\mathfrak{g}} = {\mathfrak{sl}}_2$|⁠. In this case, if we identify integral dominant weights with |${\mathbb{N}}$|⁠, Cherednik proved the equality \begin{equation}\label{eq_specialization_intro} P_{n,r,s}(q,-q^2) = J_n(K_{r,s};q) \end{equation} (1.1) where the right-hand side is the |$n{\textrm{th}}$| colored Jones polynomial of |$K_{r,s}$|⁠. (This equality is stated in our normalization conventions—the precise conversion is stated in Remark 4.3). In this article, we extend the construction of the polynomials |$P_{n,r,s}(q,t)$| from torus knots to all iterated cables of the unknot and show that these new polynomials specialize to the colored Jones polynomials. To do this, we provide a cabling formula that expresses the colored Jones polynomials of the cable |$K_{r,s}$| of a knot |$K\subset S^3$| in terms of those of |$K$|⁠. Various versions of this formula are well known and have appeared in the literature, but for completeness we prove skein-theoretic versions which are suited for our purposes in Section 2.1.5. We then show that the |$t=-q^2$| specialization of the formula defining the polynomial |$P_{n}(r,s;q,t)$| is identical to the cabling formula when |$K$| is the unknot (in this case the cable |$K_{r,s}$| is the |$(r,s)$| torus knot). We remark that the formula defining the polynomials |$P_n(r,s;q,t)$| (and our formula generalizing this to iterated torus knots) uses several structures associated to the DAHA |$\mathrm{SH}_{q,t}$|⁠, and when this formula is identified with the skein-theoretic cabling formula, each of these structures has a natural topological interpretation. More precisely, Cherednik’s construction uses the algebra |$\mathrm{SH}_{q,t}$|⁠, an |${\rm{SL}}_2({\mathbb{Z}})$| action on |$\mathrm{SH}_{q,t}$|⁠, an action of |$\mathrm{SH}_{q,t}$| on |$P = {\mathbb{C}}[x]$| (called the polynomial representation), and a pairing |$P^{op}\otimes_{\mathrm{SH}_{q,t}} P \to {\mathbb{C}}[q^{\pm 1},t^{\pm 1}]$| (where |$P^{op}$| is the twist of |$P$| by a certain anti-automorphism). In the |$t=-q^2$| specialization the algebra |$\mathrm{SH}_{q,-q^2}$| is the Kauffman bracket skein algebra of the torus, and the |${\rm{SL}}_2({\mathbb{Z}})$| action on |$\mathrm{SH}_{q,t}$| is induced by the action of the mapping class group of the torus. The polynomial representation corresponds to the skein module of a small neighborhood of a knot, which is a module over the algebra associated to its boundary torus. Finally, Cherednik’s pairing corresponds to the decomposition of |$S^3$| into the union of two unknotted solid tori, and the element on which he evaluates the pairing corresponds to the |$(r,s)$| torus knot, embedded in the common boundary of the two solid tori and colored with a Jones–Wenzl idempotent. One advantage of using the cabling formula is that it applies to all knots. As mentioned above, this allows us to extend the definition of the polynomial |$P_{n,r,s}$| from torus knots to all iterated cables of the unknot. More precisely, given a sequence |$ {\mathbf{r}}_m = (r_1,\ldots,r_m)$| and |${\mathbf{s}}_m = (s_1,\ldots,s_m)$| with |$r_i,s_i$| relatively prime integers, we define |$K({\mathbf{r}}_1,{\mathbf{s}}_1)$| to be the |$(r_1,s_1)$|-torus knot and |$K({\mathbf{r}}_m,{\mathbf{s}}_m)$| to be the |$(r_m,s_m)$| cable of |$K({\mathbf{r}}_{m-1},{\mathbf{s}}_{m-1})$|⁠. We then define a rational function |$P_n({\mathbf{r}}_m,{\mathbf{s}}_m; q,t) \in {\mathbb{C}}(q^{\pm 1},t^{\pm 1})$| for each |$n \in {\mathbb{N}}$| (see Definition 5.7) and prove the following (see Theorem 5.8): Theorem 1.1. If |$K({\mathbf{r}},{\mathbf{s}})$| is a knot which is an iterated cable of the unknot, we have the equality \[ P_n({\mathbf{r}},{\mathbf{s}};q,t=-q^2) = J_n(K({\mathbf{r}},{\mathbf{s}}); q). \] □ If the sequences |${\mathbf{r}}_m$| and |${\mathbf{s}}_m$| are length 1, then |$K({\mathbf{r}},{\mathbf{s}})$| is a torus knot and this theorem reproduces one of Cherednik’s theorems in [14]. 1.2 Arbitrary knots It is natural to ask whether some similar construction can be used to produce two-variable polynomial knot invariants for all knots. However, it seems likely that algebraic constructions using the polynomial representation can only “see” iterated torus knots or links. As a first step in this direction, in Section 3.3.2, for any knot |$K$| we define an |$\mathrm{SH}_{q,t}$| bimodule |$\bar K_{q,t}$| using a |$t$|-modification of the Kauffman bracket skein module construction. For the unknot, we show this bimodule is isomorphic to |$\mathrm{SH}_{q,t}$| itself. We further construct a canonical quotient |$K_{q,t}$| of |$\bar K_{q,t}$| which is just a left module (see Definition 3.15), and we show that for the unknot this left module is the sign representation of |$\mathrm{SH}_{q,t}$|⁠. We also show the following (see Proposition 3.16): Theorem 1.2. The vector space |$K_{q,t}$| is knot invariant which is a left module over |$\mathrm{SH}_{q,t}$|⁠. If we specialize |$t=1$| then there is a natural surjective |$\mathrm{SH}_{q,t=1}$|-module map \begin{equation}\label{eq_introsurj} K_{q,t=1} \twoheadrightarrow K_q(S^3\setminus K). \end{equation} (1.2) □ We remark that there have been a number of recent articles providing (or conjecturing) various connections between double affine Hecke algebras and certain classes of knots (e.g., [6, 14, 17, 19, 20]). However, to the best of our knowledge, the |$\mathrm{SH}_{q,t}$|-module |$K_{q,t}$| is the only proven connection between DAHAs (with arbitrary parameters) and arbitrary knots. To extract polynomial knot invariants from the classical skein module, one uses the |${\mathbb{C}}[q^{\pm 1}]$|-linear map \[\epsilon: K_q(S^3\setminus K) \to K_q(S^3) = {\mathbb{C}}[q^{\pm 1}]\] induced by the inclusion |$S^3\setminus K \to S^3$| - the key fact here is that |$K_q(S^3)$| is isomorphic to |${\mathbb{C}}[q^{\pm 1}]$|⁠. Question 1.3. For all |$t \in {\mathbb{C}}^*$|⁠, is there a canonical evaluation map |$\bar K_{q,t} \to {\mathbb{C}}[q^{\pm 1},t^{\pm 1}]$|? □ We give a positive answer to this question for the unknot in Corollary 3.21, but for other knots this seems to be a subtle question. In particular, the proof of Corollary 3.21 uses the Poincare-Birkhoff-Witt (PBW) property for |${\mathrm{H}}_{q,t}$|⁠, which is a nontrivial fact. It is unclear whether an analogue of this PBW property can be proven for nontrivial knots. However, composing the surjection (1.2) with the evaluation map |$\epsilon$| gives a positive answer to this question when |$t=1$|⁠. One might also ask if there is a similar topological construction of the (spherical) DAHA if |${\mathfrak{g}}$| has rank greater than 1. Two versions of skein relations for |${\mathfrak{g}} = {\mathfrak{sl}}_n$| (and |$t=1$|⁠) appear in [10, 31], and skein relations for |${\mathfrak{g}}$| of rank 2 appear in [22]. However, it is not clear whether the spherical subalgebra |$\mathrm{SH}_{q,t}({\mathfrak{g}})$| is a quotient of the |${\mathfrak{g}}$|-skein module of the punctured torus, and the “generators-and-relations” approach in this paper will be more difficult for higher rank |${\mathfrak{g}}$|⁠. Some brief historical remarks are in order. After the first version of the present article appeared, Cherednik and Danilenko [15] gave a construction of certain polynomials for general |${\mathfrak{g}}$| that are conjecturally related to iterated torus knots. For |${\mathfrak{g}} = {\mathfrak{sl}}_2$| we compare their construction to ours in an Appendix. After this, Morton and the author posted [27], which proved that the |${\mathfrak{gl}}_n$| polynomials defined in [15] specialized to the |${\mathfrak{gl}}_n$| Reshetikhin–Turaev invariants for iterated torus knots. Without further argument, this theorem does not directly imply the statement for |${\mathfrak{sl}}_2$| polynomials proved in this article because the relationship between the |$q,t$| polynomials for |${\mathfrak{gl}}_2$| and |${\mathfrak{sl}}_2$| is not obvious. A summary of the contents of the article is as follows. In Section 2, we recall background material about the Kauffman bracket skein module and the double affine Hecke algebra, including two cabling formulas that are essential in later sections. In Section 3, we construct |$t$|-deformed versions of the Kauffman bracket skein module of a surface and of knot complements. We then use this to give a topological construction of Cherednik’s polynomials in Section 4, and we give a topological proof that these polynomials specialize to the colored Jones polynomials of torus knots. We give an algebraic generalization of Cherednik’s construction to iterated cables of the unknot in Section 5. Finally, in an Appendix we show that when |${\mathfrak{g}} = {\mathfrak{sl}}_2$|⁠, the polynomials defined in [15] for iterated torus knots specialize to some constant times the colored Jones polynomial. 2 Background In this section, we recall background material about Kauffman bracket skein modules and double affine Hecke algebras that will be used in the remainder of the article. 2.1 Kauffman bracket skein modules In this section, we define the Kauffman bracket skein module and recall several of its properties. In particular, we describe its relation to the colored Jones polynomials and two resulting cabling formulas. 2.1.1 Knot complements Recall that two maps |$f,g:M \to N$| of manifolds are ambiently isotopic if they are in the same orbit of the identity component of the diffeomorphism group of |$N$|⁠. This is an equivalence relation, and a knot in a 3-manifold |$M$| is the equivalence class of a smooth embedding |$K: S^1 \hookrightarrow M$|⁠. For an oriented knot |$K \subset S^3$| there is a canonical identification |$T = S^1 \times S^1 \to \partial(S^3 \setminus K)$|⁠. More precisely, let |$N_K \subset S^3$| be a closed tubular neighborhood of |$K$|⁠, and let |$N_c$| be the closure of its complement. Then the following lemma provides a unique (up to isotopy) identification of |$N_K \cap N_c$| with |$S^1\times S^1$|⁠: Lemma 2.1 ([9, Chapter 3]). There is a unique (up to isotopy) pair of simple loops (the meridian |$m$| and longitude |$l$|⁠) in |$T$| subject to the conditions (1) |$m$| is nullhomotopic in |$N_K$|⁠, (2) |$l$| is nullhomologous in |$N_c$|⁠, (3) |$m,l$| intersect once in |$T$|⁠, (4) in |$S^3$|⁠, the linking numbers |$(m,K)$| and |$(l,K)$| are 1 and 0, respectively. □ 2.1.2 Kauffman bracket skein modules A framed link is an embedding of a disjoint union of annuli |$S^1 \times [0,1]$| into an oriented 3-manifold |$M$|⁠. (The framing refers to the |$[0,1]$| factor and is a technical detail that will be suppressed when possible.) We will consider framed links to be equivalent if they are ambiently isotopic. Let |$\mathscr L(M)$| be the vector space spanned by the set of ambient isotopy classes of framed unoriented links in |$M$| (including the empty link). Let |$\mathscr L'(M)$| be the smallest subspace of |$\mathscr L(M)$| containing the skein expressions |$L_+ - qL_0 - q^{-1}L_\infty$| and |$L \sqcup \bigcirc + (q^2+q^{-2})L$|⁠. The links |$L_+$|⁠, |$L_0$|⁠, and |$L_\infty$| are identical outside of a small 3-ball (embedded as an oriented sub-manifold of |$M$|⁠), and inside the 3-ball they appear as in Figure 1. (All pictures drawn in this article will have blackboard framing. In other words, a line on the page represents a strip |$[0,1]\times [0,1]$| in a tubular neighborhood of the page, and the strip is always perpendicular to the article.) Fig. 1. View largeDownload slide Kauffman bracket skein relations. Fig. 1. View largeDownload slide Kauffman bracket skein relations. Remark 2.2. Our constant |$q$| is the same as the constant |$A$| that is more commonly used in skein theory (e.g., in [7]). (We made this notational choice since we reserve the notation |$A$| for an algebra.) □ Definition 2.3 ([28]). The Kauffman bracket skein module is the vector space |$K_q(M) := \mathscr L / \mathscr L'$|⁠. It contains a canonical element |$\varnothing \in K_q(M)$| corresponding to the empty link. □ Remark 2.4. To shorten the notation, if |$M = F \times [0,1]$| for a surface |$F$|⁠, we will often write |$K_q(F)$| for the skein module |$K_q(F\times [0,1])$|⁠. □ Example 2.5. One original motivation for defining |$K_q(M)$| is the isomorphism of vector spaces \[ {\mathbb{C}} \stackrel \sim \to K_q(S^3), \quad 1 \mapsto \varnothing \] Kauffman proved that this map is an isomorphism and that the inverse image of a link is the Jones polynomial of the link. (More precisely, the image of the link is a number in |${\mathbb{C}}$| that depends polynomially on |$q \in {\mathbb{C}}^*$|⁠, and this (Laurent) polynomial is the Jones polynomial of the link, up to a normalization.) The skein relations in Figure 1 can be used to remove crossings and trivial loops of a diagram of a link until the diagram is a multiple of the empty link, which shows that the vector space map |${\mathbb{C}} \to K_q(S^3)$| sending |$\alpha \mapsto \alpha\cdot \varnothing$| is surjective. Showing it is injective is equivalent to showing the Jones polynomial of a link is well-defined. □ In general |$K_q(M)$| is just a vector space—however, if |$M$| has extra structure, then |$K_q(M)$| also has extra structure. In particular, (1) If |$M = F \times [0,1]$| for some surface |$F$|⁠, then |$K_q(M)$| is an algebra (which is typically noncommutative). The multiplication is given by “stacking links.” (2) If |$M$| is a manifold with boundary, then |$K_q(M)$| is a module over |$K_q(\partial M)$|⁠. The action is given by “pushing links from the boundary into the manifold.” (3) An oriented embedding |$M \hookrightarrow N$| of 3-manifolds induces a linear map |$K_q(M) \to K_q(N)$|⁠. (4) If |$q=\pm 1$|⁠, then |$K_q(M)$| is a commutative algebra (for any oriented 3-manifold |$M$|⁠). The multiplication is given by “disjoint union of links,” which is well defined because when |$q=\pm 1$|⁠, the skein relations allow strands to “pass through” each other. Remark 2.6. The third property may be interpreted as follows: let |$C$| be the category whose objects are oriented three-dimensional manifolds and whose morphisms are oriented embeddings. Then |$K_q(-)$| is a functor from |$C$| to the category of vector spaces. (To be pedantic, |$K_q(-)$| is functorial with respect to maps |$M \to N$| that are oriented embeddings when restricted to the interior of |$M$|⁠. In particular, if we identify a surface |$F$| with a boundary component of |$M$| and |$N$|⁠, then the gluing map |$M \sqcup N \to M \sqcup_F N$| induces a linear map |$K_q(M) \otimes_{\mathbb{C}} K_q(N) \to K_q(M\sqcup_F N)$|⁠.) We also remark that the first two properties are a special case of the third. For example, there is an obvious map |$F\times [0,1] \sqcup F\times [0,1] \to F\times [0,1]$|⁠, and the product structure of |$K_q(F \times [0,1])$| comes from the application of the functor |$K_q(-)$| to this map. □ Example 2.7. Let |$M = (S^1 \times [0,1]) \times [0,1]$| be the solid torus. The skein relations can be applied to remove crossings and trivial loops in a diagram of any link, and the result is a sum of unions of parallel copies of the loop |$u$| generating |$\pi_1(M)$|⁠. This shows that algebra map |${\mathbb{C}}[u] \to K_q(S^1\times[0,1])$| sending |$u^n$| to |$n$| parallel copies of |$u$| is surjective, and it follows from [32] that this map is injective. □ 2.1.3 The Kauffman bracket skein module of the torus We recall that the quantum torus is the algebra \[ A_q := \frac{{\mathbb{C}}\langle X^{\pm 1},Y^{\pm 1}\rangle}{XY-q^2YX}, \] where |$q \in {\mathbb{C}}^*$| is a parameter. There is a |${\mathbb{Z}}_2$| action by algebra automorphisms on |$A_q$| where the generator simultaneously inverts |$X$| and |$Y$|⁠. We define |$e_{r,s} = q^{-rs}X^{r}Y^s \in A_q$|⁠, which form a linear basis for the quantum torus |$A_q$| and satisfy the relations \[e_{r,s}e_{u,v} = q^{rv-us}e_{r+u,s+v}.\] In this section, we recall a beautiful theorem of Frohman and Gelca [18] that gives a connection between skein modules and the invariant subalgebra |$A_q^{{\mathbb{Z}}_2}$|⁠. First we establish some notation. Let |$T_n \in {\mathbb{C}}[x]$| be the Chebyshev polynomials defined by |$T_0 = 2$|⁠, |$T_1 = x$|⁠, and the relation |$T_{n+1} = xT_n-T_{n-1}$|⁠. If |$m,l$| are relatively prime, write |$(m,l)$| for the |$m,l$| curve on the torus (which is the simple curve wrapping around the torus |$l$| times in the longitudinal direction and |$m$| times in the meridian’s direction). It is clear that the links |$(m,l)^n$| span |$K_q(T^2)$|⁠, and it follows from [32] that this set is a basis. However, a more convenient basis is given by the elements |$(m,l)_T = T_d((\frac m {d}, \frac l {d}))$| (where |$d = \mathrm{gcd}(m,l)$|⁠). (We point out that since we are considering unoriented curves, |$(m,l) = (-m,-l)$|⁠.) Theorem 2.8 ([18]). The map |$f:K_q(T^2) \to A_q^{{\mathbb{Z}}_2}$| given by |$f((m,l)_T) = e_{m,l}+e_{-m,-l}$| is an isomorphism of algebras. □ (This theorem also follows from combining the results of [8] with Theorem 2.27.) Remark 2.9. From the discussion in Section 2.1.1, if |$K$| is an oriented knot, then there is a canonical identification of |$S^1\times S^1$| with the boundary of |$S^3\setminus K$|⁠, so |$K_q(S^3\setminus K)$| has a canonical |$A_q^{{\mathbb{Z}}_2}$|-module structure. (In fact, this module structure does not depend on the orientation of |$K$|⁠, but we do not need this fact.) □ 2.1.4 A topological pairing and the colored Jones polynomials Let |$K \subset S^3$| be a knot. If we identify the solid torus |$D^2\times S^1$| with a neighborhood of |$K$|⁠, then the embeddings |$D^2\times S^1 \hookrightarrow S^3$| and |$S^3\setminus K\hookrightarrow S^3$| induce a |${\mathbb{C}}$|-linear map \begin{equation}\label{eq_prepairing} K_q(D^2\times S^1)\otimes_{\mathbb{C}} K_q(S^3\setminus K) \to {\mathbb{C}}. \end{equation} (2.1) If |$\alpha$| is a link which is parallel to the boundary of |$D^2\times S^1$|⁠, it can be isotoped to a link inside |$D^2\times S^1$| or a link inside |$S^3\setminus K$|⁠, and inside |$S^3$| both these links are isotopic. Therefore, the map (2.1) descends to a pairing \begin{equation}\label{knotpairing} \langle -,-\rangle:K_q(D^2\times S^1)\otimes_{K_q(T^2)} K_q(S^3\setminus K) \to K_q(S^3) = {\mathbb{C}}. \end{equation} (2.2) The colored Jones polynomials |$J_{n}(K;q) \in {\mathbb{C}}[q^{\pm 1}]$| of a knot |$K \subset S^3$| were originally defined by Reshetikhin and Turaev [29] using the representation theory of |${\mathcal{U}}_q({\mathfrak{sl}}_2)$|⁠. (In fact, their definition works for any semisimple Lie algebra |$\mathfrak g$|⁠, but we only deal with |$\mathfrak g = {\mathfrak{sl}}_2$|⁠.) Here we recall a theorem of Kirby and Melvin that shows that |$J_{n}(K;q)$| can be computed in terms of the pairing (2.2). Let |$S_n \in {\mathbb{C}}[u]$| be the Chebyshev polynomials of the second kind, which satisfy the initial conditions |$S_0 = 1$| and |$S_1 = u$|⁠, and the recursion relation |$S_{n+1} = uS_n - S_{n-1}$|⁠. Theorem 2.10 ([21]). If |$\varnothing \in K_q(S^3\setminus K)$| is the empty link, and |$y \in K_q(T^2)$| is the longitude, then \[ J_{n}(K;q) = \langle \varnothing \cdot S_{n-1}(y), \varnothing \rangle. \] □ Remark 2.11. We remark that we avoid a common sign correction - in particular, for us, |$J_{n}(\mathrm{unknot};q) = (-1)^{n-1}(q^{2n}-q^{-2n})/(q^2-q^{-2})$|⁠. Also, with this normalization, |$J_{0}(K;q) = 0$| and |$J_{1}(K;q) = 1$| for every knot |$K$|⁠. Up to a sign, these conventions agree with the convention of labeling irreducible representations of |${\mathcal{U}}_q({\mathfrak{sl}}_2)$| by their dimension. (The |$S_{n-1}$| are the characters of these irreducible representations). □ 2.1.5 Two cabling formulas for colored Jones polynomials In this section, we describe how the colored Jones polynomials of a cable of a knot |$K$| can be computed from the skein module |$K_q(S^3\setminus K)$|⁠. There are two standard ways of defining the |$(r,s)$| cable of a knot, which we refer to as the topological and algebraic cablings, and each has its own cabling formula. Remark 2.12. We make no claim to originality for the cabling formulas we give here - they are well known and have appeared in numerous places, including [18, Theorem 7.1], [24, 34, 35]. For the sake of completeness and self-containment we will provide precise statements. □ Topological cabling Definition 2.13. Let |$K \subset S^3$| be a framed knot with 0 framing and let |$r,s \in {\mathbb{Z}}$| be relatively prime. Identify |$S^1 \times S^1$| with the boundary of a neighborhood of |$K$| such that the first copy of |$S^1$| is a meridian of |$K$| and the second is the longitude determined by the framing. (We note that since |$K$| has 0 framing, its longitude is the same as the (unique) longitude given by Lemma 2.1). Then the |$(r,s)$|topological cable|$K^{\mathrm{top}}_{r,s}$| of |$K$| is the knot which is the image of the |$(r,s)$|-curve on |$T$| and which is given the 0 framing. □ Let |$\gamma_{r,s} \in {\rm{SL}}_2({\mathbb{Z}})$| satisfy |$\gamma(0,1) = (r,s)$|⁠. The mapping class group of the torus |$T$| is |${\rm{SL}}_2({\mathbb{Z}})$|⁠, and this induces an action of |${\rm{SL}}_2({\mathbb{Z}})$| on |$K_q(T)$|⁠. By construction, we have |$\gamma_{r,s}(y) = (r,s)_T$|⁠, where |$(r,s)_T$| is the |$(r,s)$| curve on the torus |$T$| and |$y$| is the longitude of |$K$|⁠. We remark that the framing of |$(r,s)_T$| is parallel to the torus |$T$|⁠, and in particular the knot |$(r,s)_T$| is not 0-framed in |$S^3$|⁠. Definition 2.14. To give a precise statement, we give names to neighborhoods of |$K$|⁠, its boundary torus, and its cable. (1) Let |$N_{r,s}$| be a neighborhood of |$K^{\mathrm{top}}_{r,s}$|⁠, and identify |$K_q(N_{r,s}) \cong {\mathbb{C}}[u]$| (as algebras) by setting the generator |$u$| to be equal to the 0-framed knot |$K^{\mathrm{top}}_{r,s}$|⁠. (2) Let |$N_T$| be a neighborhood of |$T$|⁠. Identify |$A_q^{{\mathbb{Z}}_2}$| with |$K_q(N_T)$| by identifying |$x = X+X^{-1}$| with the meridian and |$y = Y+Y^{-1}$| with the topological longitude of Lemma 2.1. Since |$K$| is 0-framed, the longitude |$y$| is the same as the longitude determined by the framing of |$K$|⁠. (3) Let |$N_K$| be a neighborhood of |$K$|⁠, and identify |$K_q(N_K) \cong {\mathbb{C}}[y]$| (as algebras) by equating |$y$| with the 0-framed knot |$K$|⁠. This is notationally consistent because the knot |$K$| is framed parallel to |$T$|⁠, so it is the element |$Y+Y^{-1} \in A_q^{{\mathbb{Z}}_2}$| that we always call |$y$|⁠. □ The following tautological inclusions are illustrated in Figure 2: \begin{equation}\label{eq_iotatop} N_{r,s} \stackrel{\iota}{\hookrightarrow} N_T \stackrel \mu \hookrightarrow N_K. \end{equation} (2.3) Fig. 2. View largeDownload slide A cross-section of the neighborhoods |$N_{r,s}$| and |$N_T$| for |$(r,s) = (0,1)$|⁠. The outer circle bounds |$N_K$|⁠. Fig. 2. View largeDownload slide A cross-section of the neighborhoods |$N_{r,s}$| and |$N_T$| for |$(r,s) = (0,1)$|⁠. The outer circle bounds |$N_K$|⁠. We will write \begin{equation}\label{def_gammatop} \Gamma^{\mathrm{top}}_{r,s} := \mu \circ \iota \end{equation} (2.4) for the composition of these inclusions. By functoriality of skein modules and the identifications above, the map |$\Gamma^{\mathrm{top}}_{r,s}$| induces a |${\mathbb{C}}$|-linear map |$\Gamma^{\mathrm{top}}_{r,s}:{\mathbb{C}}[u] \to {\mathbb{C}}[y]$|⁠. Lemma 2.15. The induced |${\mathbb{C}}$|-linear map |$\Gamma^{\mathrm{top}}_{r,s}: {\mathbb{C}}[u] \to {\mathbb{C}}[y]$| is given by \[ \Gamma^{\mathrm{top}}_{r,s}(S_{n-1}(u)) = (-q)^{rs(n^2-1)}1_y\cdot \gamma_{r,s}(S_{n-1}(y)), \] where |$1_y \in {\mathbb{C}}[y]$| is the empty link in the skein module of a neighborhood of |$K$|⁠. □ Proof. We first compute the image |$\iota(u)$|⁠. By definition, |$\iota(u)$| is the |$r,s$| curve on |$T$|⁠, which is given the |$0$|-framing in |$S^3$|⁠. The element |$\gamma_{r,s}(y) \in K_q(T)$| is, by definition, the same curve, but its framing is parallel to |$T$|⁠. Now, the first sentence of the second paragraph of [25, p. 323] says that since |$K$| has writhe 0 (i.e., framing 0), that the framing of |$\gamma_{r,s}(y)$| in |$S^3$| is |$rs$|⁠. Therefore, |$\iota(u)$| should be twisted by |$rs$| units of framing to be isotopic to |$\gamma_{r,s}(y)$|⁠. Now |$S_{n-1}(u)$| is equal to the insertion of the |$n\textrm{th}$| Jones-Wenzl idempotent on |$u$|⁠, which means that |$\iota(S_{n-1}(u))$| is equal to the |$n{\textrm{th}}$| Jones-Wenzl idempotent inserted on the framed curve |$\gamma_{r,s}(y)$|⁠, which is then twisted by |$rs$| full twists. Since the Jones–Wenzl idempotent annihilates cups and caps, the |$rs$| full twists simplify (under the skein relations) to the displayed power of |$-q$|⁠. (For a reference for this final statement, see the first equation of the first diagram in the proof of [23, Theorem 3].) This shows |$\iota(S_{n-1}(u)) = (-q)^{rs(n^2-1)}\gamma_{r,s}(S_{n-1}(y))$|⁠. Next, |$K_q(N_K) = {\mathbb{C}}[y]$| is a right module over |$K_q(N_T)$|⁠. By the definition of this module structure, the linear map |$\mu: K_q(N_T) \to K_q(N_K)$| induced from |$\mu: N_T \to N_K$| is given by |$\mu(a) = \varnothing \cdot a$|⁠, where |$\varnothing$| is the empty link. Under the identification |${\mathbb{C}}[y] = K_q(N_k)$| the empty link corresponds to |$1$|⁠, which shows that |$\mu(a) = 1\cdot a$|⁠. This completes the proof. ■ Corollary 2.16. The colored Jones polynomials of the cable |$K_{r,s}^{\mathrm{top}}$| are computed by the formula \[ J_{n}( K_{r,s}^{\mathrm{top}};q) = (-q)^{rs(n^2-1)}\langle \varnothing \cdot \gamma_{r,s}(S_{n-1}(y)), \varnothing \rangle_K, \] where the pairing |$\langle -,-\rangle_K$| is the pairing associated to the knot |$K$| described in (2.2). □ Proof. There are two tautological inclusions |$\iota_{r,s}: N_{r,s} \to S^3$| and |$\iota_K: N_K \to S^3$|⁠. These are related to |$\Gamma_{r,s}^{\mathrm{top}}$| via the formula |$\iota_{r,s} = \iota_K \circ \Gamma_{r,s}^{\mathrm{top}}$|⁠. These inclusions all induce |${\mathbb{C}}$|-linear maps on skein modules, and we will abuse notation and denote the induced maps with the same notation. Let |$\langle -,-\rangle_{r,s}: K_q(N_{r,s}) \otimes K_q(S^3\setminus K_{r,s}^{\mathrm{top}}) \to {\mathbb{C}}[q^{\pm 1}]$| and |$\langle -,-\rangle_K: K_q(N_K) \otimes K_q(S^3\setminus K) \to {\mathbb{C}}[q^{\pm 1}]$| be the pairings described in (2.2). By definition, the evaluation |$\langle f(u), \varnothing\rangle_{r,s}$| is equal to |$\iota_{r,s}(\,f(u)) \in K_q(S^3) = {\mathbb{C}}[q^{\pm 1}]$|⁠. Similarly, |$\langle f(y),\varnothing\rangle_K = \iota_K(f(y))$|⁠. Now if we combine these statements with Theorem 2.10 and Lemma 2.15, we obtain \begin{eqnarray*} J_n(K_{r,s}^{\mathrm{top}};q) &=& \langle S_{n-1}(u),\varnothing\rangle_{r,s} \\ &=& \iota_{r,s}(S_{n-1}(u)) \\ &=& \iota_K(\Gamma_{r,s}^{\mathrm{top}}(S_{n-1}(u))) \\ &=& (-q)^{rs(n^2-1)} \langle \varnothing \cdot \gamma_{r,s}(S_{n-1}(y)),\varnothing\rangle_K. \end{eqnarray*} The first equality follows from Theorem 2.10 because |$K_{r,s}^{\mathrm{top}}$| is 0-framed, and the last equality follows from Lemma 2.15 and the definitions of |$\iota_K$| and |$\langle -,-\rangle_K$|⁠. This completes the proof. ■ This corollary gives a formula for |$J_n(K_{r,s}^{\mathrm{top}})$| in terms of the skein module of |$K$|⁠. However, the typical cabling formula that appears in the literature gives an expression in terms of Jones polynomials of |$K$| - we next derive this from Corollary 2.16. Corollary 2.17. The colored Jones polynomials of the cable |$K_{r,s}^{\mathrm{top}}$| are given by the formula \[ J_n(K_{r,s}^{\mathrm{top}};q) = (-q)^{rs(n^2-1)}\sum_{j = (-n+1)/2}^{(n-1)/2} q^{-4rj(sj+1)} J_{2sj+1}(K; q). \] □ Proof. We need to expand |$\langle \varnothing \cdot \gamma_{r,s}(S_{n-1}(y)), \varnothing\rangle_K$| in terms of the Jones polynomials of |$K$|⁠. To do this, we will use a description of the right |$A_q^{{\mathbb{Z}}_2}$|-module |$K_q(N_K)$| from [6, Lemmas 5.4, 5.5]. Let |$M = {\mathbb{C}}[Y^{\pm 1}]$| be the right |$A_q\rtimes {\mathbb{Z}}_2$| module with action \[ f(Y)\cdot Y = Yf(Y), \quad f(Y)\cdot X = -f(q^{-2}Y),\quad f(Y)\cdot s = -f(Y^{-1}). \] By [6, Lemma 5.5], the right |$A_q^{{\mathbb{Z}}_2}$|-module |$K_q(N_K)$| is isomorphic to |$M\boldsymbol{\mathrm{e}}$|⁠, where |$\boldsymbol{\mathrm{e}} = (1+s)/2$|⁠. Because of the sign in the action of |$s$|⁠, the generator of |$M\boldsymbol{\mathrm{e}}$| is |$\delta\boldsymbol{\mathrm{e}} := (Y - Y^{-1})\boldsymbol{\mathrm{e}}$|⁠, and under this isomorphism, |$\delta$| is equal to the empty link |$\varnothing \in K_q(N_K)$|⁠. In the following computation, we use the identity |$S_{n-1}(A + A^{-1}) = \sum_{j = (-n+1)/2}^{(n-1)/2} A^{2j}$|⁠, which is a standard identity for Chebyshev polynomials. \begin{eqnarray*} \varnothing \cdot \gamma_{r,s}(S_{n-1}(Y+Y^{-1})) &=& \delta \cdot \boldsymbol{\mathrm{e}} \sum_{j = (-n+1)/2}^{(n-1)/2} \gamma_{r,s}(Y^{2j}) \boldsymbol{\mathrm{e}}\\ &=& 2Y\boldsymbol{\mathrm{e}} \cdot \sum \left[q^{-rs}X^rY^s\right]^{2j} \boldsymbol{\mathrm{e}}\\ &=& 2Y\cdot \sum q^{-4rsj^2}X^{2jr}Y^{2sj}\boldsymbol{\mathrm{e}}\\ &=& 2\cdot \sum q^{-4rsj^2} q^{-4jr} X^{2jr} Y^{2sj + 1}\boldsymbol{\mathrm{e}}\\ &=& 2\cdot \sum q^{-4jr(sj+1)}Y^{2sj+1}\boldsymbol{\mathrm{e}} \\ &=& \delta \sum q^{-4jr(sj+1)}S_{2sj}(y). \end{eqnarray*} Now since |$K$| is |$0$|-framed, we can use Theorem 2.10 and Corollary 2.16 to conclude \[ J_n(K^{\mathrm{top}}_{r,s}) = (-q)^{rs(n^2-1)} \sum_{j = (-n+1)/2}^{(n-1)/2} q^{-4rj(sj+1)} J_{2sj+1}(K;q). \] ■ Remark 2.18. Since both |$K^{\mathrm{top}}_{r,s}$| and |$K$| are |$0$|-framed, the equality in the final corollary is exact (up to an overall sign |$(-1)^n$|⁠), and is not just true up to a power of |$q$|⁠. We also remark that under the substitutions |$q^{-4}_{ours} = q_{theirs}$| and |$n_{ours} = 1+b_{theirs}$|⁠, this last corollary agrees with [15, Equation (4.31)], up to an overall sign. Iterating this formula also gives a formula which is exact up to overall sign because |$2sj+1$| has the same parity for any |$j$|⁠. □ Algebraic cabling The algebraic cabling procedure is similar to the topological one, but framing is dealt with differently. We will give a more abbreviated discussion that highlights the differences. Definition 2.19. Let |$K \subset S^3$| be a framed knot with any framing and let |$r,s \in {\mathbb{Z}}$| be relatively prime. Identify |$S^1 \times S^1$| with the boundary of a neighborhood of |$K$| such that the first copy of |$S^1$| is a meridian of |$K$| and the second is the longitude determined by the framing. (We note that since |$K$| has any framing, its longitude is not necessarily the same as the (unique) longitude given by Lemma 2.1). Then the |$(r,s)$|algebraic cable|$K^{\mathrm{alg}}_{r,s}$| of |$K$| is the knot which is the image of the |$(r,s)$|-curve on |$T$| and which is given the framing parallel to the torus|$T$|⁠. □ Let |$\gamma_{r,s} \in {\rm{SL}}_2({\mathbb{Z}})$| satisfy |$\gamma(0,1) = (r,s)$|⁠. The mapping class group of the torus |$T$| is |${\rm{SL}}_2({\mathbb{Z}})$|⁠, and this induces an action of |${\rm{SL}}_2({\mathbb{Z}})$| on |$K_q(T)$|⁠. By construction, we have |$\gamma_{r,s}(y) = (r,s)_T$|⁠, where |$(r,s)_T$| is the |$(r,s)$| curve on the torus |$T$| and |$y$| is the longitude of |$K$|⁠. We remark that the framing of |$(r,s)_T$| is parallel to the torus |$T$|⁠, so |$(r,s)_T$| is isotopic to |$K^{\mathrm{alg}}_{r,s}$| as a framed knot. Definition 2.20. To give a precise statement, we give names to neighborhoods of |$K$|⁠, its boundary torus, and its cable. (1) Let |$N_{r,s}$| be a neighborhood of |$K^{\mathrm{alg}}_{r,s}$|⁠, and identify |$K_q(N_{r,s}) \cong {\mathbb{C}}[u]$| (as algebras) by setting the generator |$u$| to be equal to the framed knot |$K^{\mathrm{alg}}_{r,s}$| (which is not|$0$|-framed). (2) Let |$N_T$| be a neighborhood of |$T$|⁠. Identify |$A_q^{{\mathbb{Z}}_2}$| with |$K_q(N_T)$| by identifying |$x = X+X^{-1}$| with the meridian and |$y = Y+Y^{-1}$| with the longitude in |$T$| which is given by the framing of |$K$|⁠. (This is not the topological longitude given by Lemma 2.1.) (3) Let |$N_K$| be a neighborhood of |$K$|⁠, and identify |$K_q(N_K) \cong {\mathbb{C}}[y]$| (as algebras) by equating |$y$| with the framed knot |$K$|⁠. This means that the framing of the knot |$K$| is parallel to the torus |$T$|⁠. (This is notationally consistent as before.) □ The following tautological inclusions still hold: \[ N_{r,s} \stackrel{\iota}{\hookrightarrow} N_T \stackrel \mu \hookrightarrow N_K. \] We will write \begin{equation}\label{def_gammaalg} \Gamma^{\mathrm{alg}}_{r,s} := \mu \circ \iota \end{equation} (2.5) for the composition of these inclusions. By functoriality of skein modules and the identifications above, the map |$\Gamma^{\mathrm{alg}}_{r,s}$| induces a |${\mathbb{C}}$|-linear map |$\Gamma^{\mathrm{alg}}_{r,s}:{\mathbb{C}}[u] \to {\mathbb{C}}[y]$|⁠. Lemma 2.21. The induced |${\mathbb{C}}$|-linear map |$\Gamma^{\mathrm{alg}}_{r,s}: {\mathbb{C}}[u] \to {\mathbb{C}}[y]$| is given by \[ \Gamma^{\mathrm{alg}}_{r,s}(S_{n-1}(u)) = 1\cdot \gamma_{r,s}(S_{n-1}(y)). \] □ Proof. We first compute the image |$\iota(u)$|⁠. By definition, |$\iota(u)$| is the |$r,s$| curve on |$T$| which is given the framing parallel to |$T$|⁠. As remarked above, the element |$\gamma_{r,s}(y) \in K_q(T)$| is isotopic to |$\iota(u)$|⁠. This shows that |$\iota(S_{n-1}(u)) = \gamma_{r,s}(S_{n-1}(y))$|⁠. The computation of the map |$\mu$| is identical to the same computation in Lemma 2.15. ■ Corollary 2.22. The colored Jones polynomials of the cable |$K_{r,s}^{\mathrm{alg}}$| are computed by the formula \[ J_{n}( K_{r,s}^{\mathrm{alg}};q) = (-q)^\bullet\langle \varnothing \cdot \gamma_{r,s}(S_{n-1}(y)), \varnothing \rangle_K, \] where the pairing |$\langle -,-\rangle_K$| is the pairing associated to the knot |$K$| described in (2.2). The exponent in |$(-q)^\bullet$| depends on |$n$|⁠, |$r$|⁠, |$s$| and the framing of |$K$|⁠. □ Proof. The only difference between the proof of this and the proof of Corollary 2.16 is that the knot |$K_{r,s}^{\mathrm{alg}}$| is not |$0$|-framed. However, since the Jones–Wenzl idempotent kills cups and caps, the evaluation |$\langle S_{n-1}(u),\varnothing\rangle_{r,s}$| will be a power of |$-q$| times the Jones polynomial |$J_{n}(K_{r,s}^{\mathrm{alg}};q)$|⁠. ■ 2.2 The |$\boldsymbol{{\mathfrak{sl}}_2}$| double affine Hecke algebra In this section, we recall background about the double affine Hecke algebra |${\mathrm{H}}_{q,t}$| of type |$A_1$| that is required for Cherednik’s construction and for our purposes later. The standard reference for the material in this section is [12]. 2.2.1 The Poincarè–Birkhoff–Witt property We first give a presentation of the algebra |${\mathrm{H}}_{q,t}$|⁠. Definition 2.23. Let |${\mathrm{H}}_{q, t}$| be the algebra generated by |$X^{\pm 1}$|⁠, |$Y^{\pm 1}$|⁠, and |$T$| subject to the relations \begin{equation} \label{daha} TXT=X^{-1},\quad TY^{-1}T = Y, \quad XY=q^2YXT^2, \quad (T-t)(T+t^{-1})=0. \end{equation} (2.6) □ We remark that we have replaced the |$q$| that is standard in the third relation with |$q^2$| to agree with the standard conventions for the skein relations in Figure 1. Also, the fourth relation implies that |$T$| is invertible, with inverse |$T^{-1} = T + t^{-1} - t$|⁠. Finally, if we set |$t=1$|⁠, then the fourth relation reduces to |$T^2 = 1$|⁠, and the third relation becomes |$XY=q^2YX$|⁠. These imply that |${\mathrm{H}}_{q,1}$| is isomorphic to the cross product |$A_q\rtimes {\mathbb{Z}}_2$| (where the generator of |${\mathbb{Z}}_2$| acts by inverting |$X$| and |$Y$|⁠). One of the key properties of |$ {\mathrm{H}}_{q,t} $| is the so-called PBW property, which says that, for all |$\,q,t \in {\mathbb{C}}$|⁠, the multiplication map yields a linear isomorphism $$ {\mathbb{C}}[X^{\pm 1}] \otimes {\mathbb{C}}[{\mathbb{Z}}_2] \otimes {\mathbb{C}}[Y^{\pm 1}] \stackrel{\sim}{\to} {\mathrm{H}}_{q,t}. $$ Another way of stating this property is that the elements |$\,\{X^n T^\varepsilon Y^m\, :\, m,n \in {\mathbb{Z}}\,,\,\varepsilon = 0,\,1\}\,$| form a linear basis in |$ {\mathrm{H}}_{q,t} $|⁠. (See [12], Theorem 2.5.6(a).) 2.2.2 The spherical subalgebra If |$ t \ne \pm i $|⁠, the algebra |$ {\mathrm{H}}_{q,t} $| contains the idempotent |$\boldsymbol{\mathrm{e}} := (T+t^{-1})/(t+t^{-1})$| (the identity |$\,\boldsymbol{\mathrm{e}}^2 = \boldsymbol{\mathrm{e}} \,$| is equivalent to the last relation in (2.6)). The spherical subalgebra of |$ {\mathrm{H}}_{q,t} $| is \begin{equation} \label{salg} \mathrm{SH}_{q,t} := \boldsymbol{\mathrm{e}}{\mathrm{H}}_{q,t}\boldsymbol{\mathrm{e}} . \end{equation} (2.7) Note that |$ {\mathrm{SH}_{q,t}}$| inherits its additive and multiplicative structure from |$ {\mathrm{H}}_{q,t} $|⁠, but the identity element of |${\mathrm{SH}_{q,t}}$| is |$ \boldsymbol{\mathrm{e}} $|⁠, which is different from |$\, 1 \in {\mathrm{H}}_{q,t} $|⁠. If |$t^2q^{-2} - t^{-2}q^2$| is invertible, then the next lemma shows that |$ {\mathrm{SH}_{q,t}} $| is Morita equivalent to |$ {\mathrm{H}}_{q,t} $|⁠; the mutually inverse equivalences are given by \begin{equation} \label{mor} {\tt{Mod}}\,{\mathrm{H}}_{q,t} \to {\tt{Mod}}\,{\mathrm{SH}_{q,t}}\, ,\ M \mapsto \boldsymbol{\mathrm{e}} M; \qquad {\tt{Mod}}\,{\mathrm{SH}_{q,t}} \to {\tt{Mod}}\,{\mathrm{H}}_{q,t}\, ,\ M \mapsto {\mathrm{H}}_{q,t}\,\boldsymbol{\mathrm{e}} \otimes_{{\mathrm{SH}_{q,t}}} M. \end{equation} (2.8) Lemma 2.24. If |$t^2q^{-2}-t^{-2}q^2$| is invertible, then |${\mathrm{H}}_{q,t}\boldsymbol{\mathrm{e}} {\mathrm{H}}_{q,t} = {\mathrm{H}}_{q,t}$|⁠. □ Proof. Define |$a := t^{-1}X-tX^{-1}$| and |$\pi := YT^{-1}$|⁠. The statement then follows from the fact that |$X\pi$| is invertible and from the following identities: \begin{align*} \pi^2 &= 1\\ \pi X &= q^2 X^{-1} \pi\\ a &=X\boldsymbol{\mathrm{e}} - \boldsymbol{\mathrm{e}} X^{-1} \\ \pi a + t^{-2}q^2 a\pi &= -t^{-1}(t^2q^{-2}-t^{-2}q^2)X\pi. \end{align*} More precisely, let |$I$| be the two-sided ideal generated by |$\boldsymbol{\mathrm{e}}$|⁠. Then the third identity shows |$a$| is in |$I$|⁠, the final identity follows from the first two, and if |$(t^2q^{-2}-t^{-2}q^2)$| is a unit then |$I$| contains a unit because of the final identity. ■ In the case |$t=1$|⁠, there is an isomorphism |$A_q^{{\mathbb{Z}}_2} \cong \mathrm{SH}_{q,1}$| given by |$w \mapsto \boldsymbol{\mathrm{e}} \bar w \boldsymbol{\mathrm{e}}$|⁠, where |$w \in A_q^{{\mathbb{Z}}_2}$| is a symmetric word in |$X,Y$|⁠, and |$\bar w$| is the same word, viewed as an element of |${\mathrm{H}}_{q,t}$|⁠. For later use we will need a presentation of |$\mathrm{SH}_{q,t}$| which we give here. Definition 2.25. Let |$B'_q$| be the algebra generated by |$x,y,z$| modulo the following relations: \begin{equation}\label{relationsforB'} [x,y]_q = (q^2-q^{-2})z,\quad [z,x]_q = (q^2-q^{-2})y,\quad [y,z]_q = (q^2-q^{-2})x. \end{equation} (2.9) Also, define |$B_{q,t}$| to be the quotient of |$B'_q$| by the additional relation \begin{equation}\label{casimir_rel} q^2x^2 + q^{-2}y^2+ q^2z^2 -qxyz= \left( \frac t q - \frac q t\right)^2 + \left( q + \frac 1 q\right)^2. \end{equation} (2.10) □ Remark 2.26. The element on the left-hand side of (2.10) is central in |$B'_q$| (see Corollary 3.7), so |$B_q$| is the quotient of |$B'_q$| by a central character. □ Theorem 2.27 ([33]). There is an algebra isomorphism |$f:B_{q,t} \to \mathrm{SH}_{q,t}$| defined by the following formulas: \begin{align}\label{xyz} x &\mapsto (X+X^{-1})\boldsymbol{\mathrm{e}}\notag\\ y &\mapsto (Y+Y^{-1})\boldsymbol{\mathrm{e}}\\ z &\mapsto q^{-1}(XYT^{-2}+X^{-1}Y^{-1}) \boldsymbol{\mathrm{e}}.\notag \end{align} (2.11) □ Proof. The fact that (2.11) gives a well-defined algebra map can be checked directly, and the fact that it is an isomorphism is proved in [33]. (See also [6, Theorem 2.20] for the precise conversion between Terwilliger’s notation and ours.) ■ Remark 2.28. A priori, it is not obvious that the elements on the right-hand side of (2.11) are contained in |$\boldsymbol{\mathrm{e}} {\mathrm{H}}_{q,t} \boldsymbol{\mathrm{e}}$|⁠. However, short computations show that if we take |$a \in {\mathrm{H}}_{q,t}$| to be either |$X+X^{-1}$|⁠, |$Y+Y^{-1}$|⁠, or |$XYT^{-2}+X^{-1}Y^{-1}$|⁠, then |$a\boldsymbol{\mathrm{e}} = \boldsymbol{\mathrm{e}} a$|⁠, and this implies |$a \boldsymbol{\mathrm{e}} = \boldsymbol{\mathrm{e}} a \boldsymbol{\mathrm{e}} \in \boldsymbol{\mathrm{e}} {\mathrm{H}}_{q,t}\boldsymbol{\mathrm{e}}$|⁠. □ For later reference, we include a lemma that is useful for establishing isomorphisms of |$B'_q$|-modules. Lemma 2.29. Suppose that |$M$| and |$N$| are modules over |$B'_q$|⁠, and that as a |${\mathbb{C}}[x]$|-module |$M$| is generated by elements |$\{m_i \in M\}$|⁠. Furthermore, suppose that |$f:M \to N$| is a morphism of |${\mathbb{C}}[x]$|-modules that satisfies |$f(ym_i) = yf(m_i)$| and |$f(zm_i) = zf(m_i)$|⁠. Then |$f$| is a morphism of |$B'_q$|-modules. □ Proof. By definition, the elements |$x,y,z \in B'_q$| satisfy the commutation relations (2.9). An arbitrary element of |$M$| can be written as |$m = \sum_{i=1}^n c_i p_i(x) m_i$|⁠, and using the |${\mathbb{C}}[x]$|-linearity of |$f$| and the commutation relations, powers of |$x$| in the expressions |$ym$| and |$zm$| can inductively be moved to the left. This shows that |$f(ym) = yf(m)$| and |$f(zm) = zf(m)$| for arbitrary |$m \in M$|⁠, which completes the proof. ■ 2.2.3 The standard and sign polynomial representation Our definition of |${\mathrm{H}}_{q,t}$| was in terms of generators and relations. However, |${\mathrm{H}}_{q,t}$| can also be viewed as a family of subalgebras of |${\rm{End}}_{\mathbb{C}}({\mathbb{C}}[X,X^{-1}])$| using the construction we describe in this section. We first introduce the following linear operators on |${\mathbb{C}}[X^{\pm 1}]$|⁠: \begin{equation} \label{op} {\hat{x}}[f(X)] := X f(X)\ ,\quad {\hat{s}}[f(X)] := f(X^{-1})\ ,\quad {\hat{y}}[f(X)] := f(q^{-2}X)\ . \end{equation} (2.12) Notice that these operators are invertible and satisfy the relations \begin{equation} \label{rop} {\hat{s}}^2 =1 \ , \quad {\hat{s}}\,{\hat{x}} = {\hat{x}}^{-1} {\hat{s}} \ , \quad {\hat{s}}\,{\hat{y}} = {\hat{y}}^{-1}{\hat{s}} \ ,\quad {\hat{x}}\,{\hat{y}} = q^2\,{\hat{y}}\,{\hat{x}}\ . \end{equation} (2.13) Thus, they define a representation of the crossed product |$A_q \rtimes {\mathbb{Z}}_2$|⁠. The action of |$ {\mathrm{H}}_{q,t} $| on |$ {\mathbb{C}}[X^{\pm 1}]$| can be described by \begin{equation} \label{du} X \mapsto {\hat{x}} \ ,\quad T \mapsto {\hat{T}} := t\cdot{\hat{s}} + \frac{t-t^{-1}}{X^2-1}({\hat{s}}-1) \ ,\quad Y \mapsto {\hat{y}}\,{\hat{s}}\,{\hat{T}}. \end{equation} (2.14) The operator |$ {\hat{T}} $| is called the Demazure–Lusztig operator (cf. [12], (1.4.26), (1.4.27)). Formally it is an operator on rational functions |${\mathbb{C}}(X)$|⁠, but it preserves the subspace |${\mathbb{C}}[X^{\pm 1}] \subset {\mathbb{C}}(X)$| because |$f(X^{-1})-f(X)$| is always divisible (in |${\mathbb{C}}[X^{\pm 1}]$|⁠) by |$X^2-1$|⁠. These assignments can be rephrased as follows: let |${\mathcal{D}}_q$| be the localization of |$A_q\rtimes {\mathbb{Z}}_2$| with respect to the multiplicative set consisting of all nonzero polynomials in |$X$|⁠. Then formulas (2.14) give an embedding \begin{equation} \label{duem} \Theta_{q,t}:\, {\mathrm{H}}_{q,t} \hookrightarrow {\mathcal{D}}_q. \end{equation} (2.15) The sign representation is defined similarly—we first define operators \begin{equation} {\hat{x}}[f(X)] := X f(X)\ ,\quad {\hat{s}}_-[f(X)] := -f(X^{-1})\ ,\quad {\hat{y}}_-[f(X)] := -f(q^{-2}X). \end{equation} (2.16) These operators give |${\mathbb{C}}[X^{\pm 1}]$| the structure of an |$A_q\rtimes{\mathbb{Z}}_2$|-module. We then define \begin{equation}\label{eq_signde} X \mapsto {\hat{x}} \ ,\quad T \mapsto {\hat{T}}_- := -t^{-1}\cdot{\hat{s}}_- + \frac{t-t^{-1}}{X^2-1}({\hat{s}}_-+1) \ ,\quad Y \mapsto {\hat{y}}_-\,{\hat{s}}_-\,{\hat{T}}_-. \end{equation} (2.17) It can be checked directly that these assignments give an embedding |$\Theta^-_{q,t}:{\mathrm{H}}_{q,t} \to {\mathcal{D}}_q$|⁠. Definition 2.30. Let |$P^+,P^- := {\mathbb{C}}[X^{\pm 1}]$| be the |${\mathrm{H}}_{q,t}$|-modules given by formulas (2.14) and (2.17), respectively. We refer to these as the polynomial representation and the sign representation. □ Remark 2.31. The modules |$P^+$| and |$P^-$| can also be viewed as induced modules as follows. Let |${\mathrm{H}}_Y \subset {\mathrm{H}}_{q,t}$| be the subalgebra generated by |$Y^{\pm 1}$| and |$T$|⁠. Then |${\mathrm{H}}_Y$| is the affine Hecke algebra (of type |$A_1$|⁠) and it has two natural one-dimensional modules, |${\mathbb{C}}_t$| and |${\mathbb{C}}_{-t^{-1}}$|⁠. On the module |${\mathbb{C}}_t$| both |$Y$| and |$T$| act by multiplication by |$t$|⁠, and on |${\mathbb{C}}_{-t^{-1}}$| they act by multiplication by |$-t^{-1}$|⁠. Then |$P^+$| and |$P^-$| are the |${\mathrm{H}}_{q,t}$|-modules which are induced from |${\mathbb{C}}_t$| and |${\mathbb{C}}_{-t^{-1}}$|⁠, respectively. □ 2.2.4 The symmetric polynomial representation Under the Morita equivalence (2.8), the |$ {\mathrm{H}}_{q,t}$|-module |$ P^+ $| corresponds to the |$ {\mathrm{SH}_{q,t}}$|-module $$ \boldsymbol{\mathrm{e}} P^+ = \boldsymbol{\mathrm{e}}{\mathrm{H}}_{q,t}/(\boldsymbol{\mathrm{e}}{\mathrm{H}}_{q,t} \, (T-t) + \boldsymbol{\mathrm{e}}{\mathrm{H}}_{q,t}\, (Y-t)) \cong \boldsymbol{\mathrm{e}}\cdot {\mathbb{C}}[X^{\pm 1}]. $$ Now, note that the subspace |$ \boldsymbol{\mathrm{e}}{\mathbb{C}}[X^{\pm 1}] \subset {\mathbb{C}}[X^{\pm 1}] $| is the image of the projector |$ \boldsymbol{\mathrm{e}} = (t^{-1}+T)/(t^{-1}+t)$| and hence the kernel of |$1 - \boldsymbol{\mathrm{e}}$|⁠. By the Bernstein–Zelevinsky lemma (see [12], p. 202), the kernel of the operator |$ T - t $| (acting on |$ {\mathbb{C}}[X^{\pm 1}] $| as in (2.14)) is exactly |${\mathbb{C}}[X^{\pm 1}]^{{\mathbb{Z}}_2} = {\mathbb{C}}[X+X^{-1}]$|⁠, the subspace of symmetric Laurent polynomials. Thus, for all parameters |$q,t \in {\mathbb{C}}^*$|⁠, the spherical algebra |$ {\mathrm{SH}_{q,t}} $| acts on |$ {\mathbb{C}}[X+X^{-1}] $| via the identification \begin{equation} \label{sphr} {\mathbb{C}}[X+X^{-1}] = \boldsymbol{\mathrm{e}}\cdot {\mathbb{C}}[X^{\pm 1}] \cong \boldsymbol{\mathrm{e}} P^+ \quad f(X+X^{-1}) \leftrightarrow \boldsymbol{\mathrm{e}} f(X+X^{-1}) \leftrightarrow [\boldsymbol{\mathrm{e}} f(X+X^{-1})]. \end{equation} (2.18) Since |$Y+Y^{-1}$| commutes with |$\boldsymbol{\mathrm{e}}$| it preserves the subspace |${\mathbb{C}}[X+X^{-1}]\subset {\mathbb{C}}[X^{\pm 1}]$|⁠. This operator is called the Macdonald operator, and it plays a fundamental role in the representation-theoretic approach to the theory of Macdonald polynomials. A computation shows that this operator can be written as \begin{equation} \label{mac} L_{q,t} := Y + Y^{-1} = \frac{tX^{-1}-t^{-1}X}{X^{-1}-X}\,{\hat{y}} + \frac{t^{-1}X^{-1}-tX}{X^{-1}-X}\,{\hat{y}}^{-1}\ . \end{equation} (2.19) Remark 2.32. Under the identification |$\mathrm{SH}_{q,t=1} \cong K_q(T^2)$| in Theorem 2.8, the Macdonald operator is identified with the longitude of the torus. □ 2.2.5 Rank 1 Macdonald polynomials We briefly review the definition of Macdonald polynomials of type |$A_1$| (a.k.a. the Rogers or |$(q,t)$|-ultraspherical polynomials, see [5]). This family of orthogonal polynomials depends on two parameters |$ q,t \in {\mathbb{C}}^*$| and forms a basis in the space |$ {\mathbb{C}}[X+X^{-1}] = {\mathbb{C}}[x] $| of symmetric Laurent polynomials. (The Macdonald polynomials of type |$A_1$| constitute a subfamily of the four-parameter family of the so-called Askey-Wilson polynomials, which is the most general family of orthogonal polynomials of one variable. The Askey-Wilson polynomials are controlled by the DAHA of type |$CC^\vee$|⁠.) These polynomials can be naturally defined for an arbitrary root system; they possess some remarkable properties which were first conjectured by I. G. Macdonald and proved by Cherednik (see [12, Sect. 1.4] for a discussion of these polynomials and the Macdonald conjectures). The Macdonald polynomials can be defined as solutions of the eigenvalue problem for the Macdonald operator: $$ L_{q,t}[\varphi] = (\lambda + \lambda^{-1})\,\varphi. $$ Theorem 2.33. There is a unique family of symmetric Laurent polynomials |$\{p_0(x),\,p_1(x),\,\ldots\} \subset {\mathbb{C}}[X^{\pm 1}]^{{\mathbb{Z}}_2}={\mathbb{C}}[x] $|⁠, satisfying \begin{align*} &(a) L_{q,t}[p_n] = (tq^{2n} + t^{-1} q^{-2n})\,p_n \ ,\\ &(b) p_0(x) = 1 \ ,\quad p_n(x) = X^n + X^{-n} + \sum_{|m| < n}\,c_m\,x^m\ ,\ n \ge 1. \end{align*} □ With this choice of normalization, the Macdonald polynomials depend rationally on |$q,t$|⁠. For example, the first members of this family are $$ p_0(x) = 1\ ,\quad p_1(x) = X + X^{-1} \ ,\quad p_2(x) = X^2 + X^{-2} + (1-t^2)(1+q^4)/(1- t^2 q^4). $$ 2.2.6 The Dunkl–Cherednik pairing Here we recall an important property of the polynomial representation which is analogous to the Shapovalov form from Lie theory. First we define an anti-automorphism of |${\mathrm{H}}_{q,t}$|⁠: \begin{equation}\label{eq_antiautomorphism} \phi: {\mathrm{H}}_{q,t} \to {\mathrm{H}}_{q,t}\ ,\quad X \mapsto Y^{-1}\ ,\quad Y \mapsto X^{-1}\ , \quad T \mapsto T\ . \end{equation} (2.20) We write |$\phi(P^+)$| for the twist of the module |$P^+$| by this anti-automorphism. Lemma 2.34. The vector space |$\phi(P^+) \otimes_{{\mathrm{H}}_{q,t}} P^+$| is one-dimensional. □ Proof. This follows from the PBW property combined with some computations (see [12, Section 1.4.2]). ■ Corollary 2.35. There are pairings \[ \langle -,-\rangle_{q,t}: \phi(P^+) \otimes_{{\mathrm{H}}_{q,t}} P^+ \to {\mathbb{C}}, \quad \langle -,-\rangle_{q,t}: \phi(P^+\boldsymbol{\mathrm{e}}) \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^+ \to {\mathbb{C}}, \] and these pairings are both uniquely determined by the condition |$\langle 1,1\rangle = 1$|⁠. □ Proof. The existence and uniqueness of the first pairing follow from Lemma 2.34. Since |$\mathrm{SH}_{q,t}$| and |${\mathrm{H}}_{q,t}$| are Morita equivalent, the natural map |$\phi(P^+\boldsymbol{\mathrm{e}}) \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^+ \to \phi(P^+) \otimes_{{\mathrm{H}}_{q,t}} P^+$| is an isomorphism of vector spaces (see, e.g., [6, Lemma 5.2]). This shows the existence and uniqueness of the second pairing. ■ Remark 2.36. This pairing is very closely related to the topological pairing (2.2). More precisely, if the construction in this section is repeated for the sign representation |$P^-$|⁠, then this pairing at |$t=1$| is exactly the same as the pairing (2.2) when |$L$| is the unknot (see Lemma 4.5). □ 2.2.7 The sign representation In this subsection, we recall some facts about the sign representation |$P^-$| of |${\mathrm{H}}_{q,t}$|⁠, which will be used in Section 3.4. The symmetric sign representation |$\boldsymbol{\mathrm{e}} P^-$| can be identified with |${\mathbb{C}}[x]$| as follows. First, we define \[ \delta_t := tX^{-1}-t^{-1}X\] and note that in the sign representation we have |$\boldsymbol{\mathrm{e}} \delta_t = \delta_t$| and |$\boldsymbol{\mathrm{e}}\cdot 1 = 0$|⁠. Since |$T$| commutes with elements of |${\mathbb{C}}[X+X^{-1}] \subset {\mathrm{H}}_{q,t}$|⁠, the |$-t^{-1}$| and |$t$| eigenspaces of |$T$| are \[ P^- = {\mathbb{C}}[X+X^{-1}] \oplus {\mathbb{C}}[X+X^{-1}]\delta_t. \] Therefore, |$\boldsymbol{\mathrm{e}}$| kills the first factor in this decomposition, and we obtain \begin{equation}\label{eq_symsigndecomp} \boldsymbol{\mathrm{e}} P^- = {\mathbb{C}}[x]\delta_t \subset {\mathbb{C}}[X^{\pm 1}], \end{equation} (2.21) where |$x = X + X^{-1}$|⁠. As in the previous section, we write |$P^-\boldsymbol{\mathrm{e}}$| for the right |${\mathrm{H}}_{q,t}$|-module that is the twist of |$\boldsymbol{\mathrm{e}} P^-$| by the anti-automorphism |$\phi$| from (2.20). Lemma 2.37. There is a well-defined pairing |$\langle -,-\rangle: P^-\boldsymbol{\mathrm{e}} \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^- \to {\mathbb{C}}$|⁠, and this pairing is uniquely determined by the condition |$\langle \delta_t, \delta_t\rangle = 1$|⁠. □ Proof. The uniqueness of the claimed pairing follows from the decomposition (2.21) and from the fact that |$\delta_t$| is an eigenvector of |$\boldsymbol{\mathrm{e}}(Y+Y^{-1})\boldsymbol{\mathrm{e}}$|⁠. To show existence, we first note that there is an isomorphism |$f: {\mathrm{H}}_{q,t} \to {\mathrm{H}}_{q,-t^{-1}}$| determined by \[ f(X) = X,\quad f(Y) = Y,\quad f(T) = T. \] The twist |$f^*(P^+)$| of the standard polynomial representation by this isomorphism is the sign representation, and combining this with Lemma 2.34 shows that |$\phi(P^-)\otimes_{{\mathrm{H}}_{q,t}} P^-$| is one-dimensional. Then the claim follows using the same argument as in Corollary 2.35. ■ We will also need the analogues of the Macdonald polynomials for the sign representation. Definition 2.38. We define the sign Macdonald polynomials by |$p_n^-(x) = p_n^-(x;q,t) := p_n(x;q,-t^{-1})$|⁠. □ Remark 2.39. By the proof of Lemma 2.37, the polynomials |$p_n^-(x)\delta_t \in \boldsymbol{\mathrm{e}} P^-$| are eigenvectors for the Macdonald operator |$Y+Y^{-1}$|⁠. □ 3 A modified Kauffman Bracket Skein Module In this section, we discuss deformations of skein modules to modules over the DAHA |${\mathrm{H}}_{q,t}$| of type |$A_1$|⁠. We use a modification of the Kauffman bracket skein module to give a topological construction of the spherical subalgebra |$\mathrm{SH}_{q,t}$|⁠. We also give a construction which associates an |$\mathrm{SH}_{q,t} $|-module to each knot in |$S^3$|⁠. 3.1 Modified KBSM for surfaces Our goal in this section is to define a 2-parameter skein algebra |$K_{q,t}(F)$| for a (connected) surface |$F$|⁠. As we describe below, the algebra |$K_{q,t}(F)$| will be a quotient of the skein module |$K_q(F \setminus \{p\})$| of the punctured surface by an ideal depending on the parameter |$t \in {\mathbb{C}}^*$|⁠. For general surfaces, there is a surjection |$K_{q,t=1}(F) \twoheadrightarrow K_q(F)$|⁠, but the specialization |$K_{q,t=1}(F)$| is “bigger” than |$K_q(F)$|⁠. More precisely, the set of links with non-crossing, nontrivial components forms a (linear) basis of |$K_q(F)$| but does not span |$K_{q,t=1}(F)$|⁠. However, if |$F$| is the torus |$T^2$|⁠, then this set is a linear basis for |$K_{q,t}(T^2)$| (for all |$t$|⁠). Therefore, |$K_{q,t}(T^2)$| can be viewed as a (flat) deformation of |$K_q(T^2)$|⁠. We will show that this deformation is the same as the algebraic deformation given by the spherical subalgebra |$\mathrm{SH}_{q,t}$| described previously. Definition 3.1. We fix a point |$p \in F$| and define the following: (1) A vertical special strand is a homeomorphism |$[0,1] \to \{p\} \times [0,1] \subset F \times [0,1]$|⁠. (2) A framed link with a vertical special strand is an isotopy class of embeddings |$[0,1] \sqcup \left(\sqcup_n S^1\times[0,1]\right)\hookrightarrow F \times [0,1]$|⁠, where |$n \geq 0$| and the isotopy class contains a representative whose embedding of |$[0,1]$| is a vertical special strand. (The allowed isotopies must fix the boundary of the 3-manifold, and in particular the endpoints of the special strand are fixed). □ Let |$\mathcal L_{q,t}$| be the vector space spanned by framed links with a vertical special strand. (We remark that |$\mathcal L_{q,t}$| is isomorphic to the space spanned by framed links in a thickening of the punctured surface. However, we phrase the definition using “special strands” because we will later want to extend this construction to 3-manifolds that are not thickened surfaces, and it seems less convenient to use the “puncture” definition in this situation.) Let |$\mathcal L'_{q,t}\subset {\mathbb{C}} \mathcal L_{q,t}$| be the subspace generated by elements of the form \begin{equation}\label{modifiedkbsm_rel} L_+-qL_0-q^{-1}L_\infty, \quad (L \sqcup \bigcirc) + (q^2 + q^{-2})L, \textrm{ and } L_s+(q^2t^{-2}+q^{-2}t^2)L', \end{equation} (3.1) where |$L_+,L_0,L_\infty$| are links that are identical outside of a ball, and inside the ball appear as the first, second, and third terms of Figure 1 (respectively). Similarly, |$L_s,L'$| are links which are identical outside a ball, and inside a ball appear as in Figure 3. (The dotted lines in Figure 3 represent the special strand, and the solid loop in |$L_s$| is a component of a framed link in the skein module.) We note that the third relation in (3.1) does not require the special strand to be framed, since the relation does not use the framing of the special strand in any way. Fig. 3. View largeDownload slide Skein relation for special strands. Fig. 3. View largeDownload slide Skein relation for special strands. Definition 3.2. The modified Kauffman bracket skein module|$K_{q,t}(F)$| is the algebra |${\mathbb{C}} \mathcal L_{q,t} / \mathcal L'_{q,t}$|⁠. □ Remark 3.3. In words, the first two relations in (3.1) say that the standard skein relations apply between links in |$F \times [0,1]$|⁠. The third relation (between |$L_s$| and |$L$|⁠) is saying “a loop around the special strand can be removed at the cost of a constant.” In other words, |$K_{q,t}(F)$| is isomorphic to |$K_{q}(F \setminus p)$| modulo the relation that says: if |$L_p$| is a horizontally-framed loop encircling the puncture |$p$|⁠, then |$L_p = -q^2t^{-2} - q^{-2}t^2$|⁠. □ The algebra structure is given by “stacking links" as before. More precisely, given two links |$L_1,L_2 \subset F \times [0,1]$|⁠, we take two copies of |$F \times [0,1]$|⁠, each containing one of the |$L_i$|⁠, and glue |$F \times \{1\}$| in one to |$F\times \{0\}$| in the other via the identity map. Since each special strand begins and ends at |$p$|⁠, the special strands glue together. The identity element of this algebra is the link with one special strand and no other components. We also remark that the identity component of the diffeomorphism group of a surface acts transitively, so different choices of |$p$| give isomorphic algebras. We therefore do not include the point |$p \in F$| in the notation. Lemma 3.4. There is a natural surjective algebra map |$K_{q,t=1}(F) \twoheadrightarrow K_{q}(F)$|⁠. □ Proof. First, there is a natural map |$f: K_q(F \setminus p) \to K_q(F)$|⁠. Let |$L_p \in K_q(F\setminus p)$| be the loop which encircles the puncture |$p$|⁠. Under this map, |$L_p$| is sent to a nullisotopic loop, which is equal to |$-q^2-q^{-2}$| by the (standard) skein relations. Second, if |$t=1$|⁠, then |$L_p = -q^2t^{-2}-q^{-2}t^2 = -q^2-q^{-2}$| in |$K_{q,t=1}(F)$|⁠. Third, by Remark 3.3, |$K_{q,t}(F)$| is the quotient of |$K_q(F\setminus p)$| by the relation |$L_p = -q^2t^{-2}-q^{-2}t^2$|⁠. Combining these three statements shows that if |$t=1$|⁠, then the map |$f: K_q(F \setminus p) \to K_q(F)$| factors through to the quotient, so induces a map |$\tilde f: K_{q,t=1}(F) \to K_q(F)$|⁠. The map |$\tilde f$| is surjective because |$f$| is. ■ 3.2 The torus We now compute the algebra structure of |$K_{q,t}(T^2)$|⁠. We first recall a useful result from [8]. Let |$T'$| be the torus with one puncture, and let |$x'$| and |$y'$| be simple closed curves that intersect once, and let |$z'$| be the (simple closed) curve with the coefficient |$q$| in the resolution of the product |$x'y'$|⁠. (See Figure 4). Fig. 4. View largeDownload slide The generators for the punctured torus. Fig. 4. View largeDownload slide The generators for the punctured torus. We also need the algebras |$B'_q$| and |$B_{q,t}$| from Section 2.2.2. We recall that |$B'_q$| is the algebra generated by |$x,y,z$| modulo the following relations: \begin{equation}\label{relationsforB'_top} [x,y]_q = (q^2-q^{-2})z,\quad [z,x]_q = (q^2-q^{-2})y,\quad [y,z]_q = (q^2-q^{-2})x \end{equation} (3.2) and that |$B_{q,t}$| is the quotient of |$B'_q$| by the additional relation \begin{equation}\label{casimir_rel_top} q^2x^2 + q^{-2}y^2+ q^2z^2 -qxyz= \left( \frac t q - \frac q t\right)^2 + \left( q + \frac 1 q\right)^2. \end{equation} (3.3) Theorem 3.5. [8, Theorem 2.1] There is an algebra isomorphism |$B'_q \to K_q(T')$| induced by the assignments |$x \mapsto x'$|⁠, |$y \mapsto y'$|⁠, and |$z \mapsto z'$|⁠. □ Remark 3.6. It is a general fact that if |$F$| is any surface and |$\partial$| is a curve parallel to a boundary component of |$F$|⁠, then |$\partial$| is a central element in |$K_q(F)$|⁠. This is true because if |$\partial$| is “on top of” a link |$L$|⁠, then |$\partial$| can be shrunk to be very close to the boundary, slid down the boundary until it is below |$L$|⁠, and then expanded. In other words, the link |$\partial L$| is isotopic to the link |$L\partial$|⁠. □ Corollary 3.7. The element |$w = q^2x^2 + q^{-2}y^2+ q^2z^2-qxyz$| is central in |$B'_q$|⁠. □ Proof. If we show that the element |$-w+q^2+q^{-2}$| is the loop parallel to the boundary of |$T'$|⁠, then the considerations in the previous remark apply to prove the claim. To see this identity we use [8, Equation (2)]. Write |$\delta$| for the loop around the puncture and |$\alpha$| for the curve with coefficient |$q^{-1}$| in Figure 4. Then in our notation their identity is \[ \alpha z = q^2 x^2 + q^{-2} y^2 - q^2 - q^{-2} + \partial. \] Then the identity |$xy = qz + q^{-1} \alpha$| of Figure 4 shows that \[ qxyz = q^2z^2 + (q^2 x^2 + q^{-2} y^2 - q^2 - q^{-2} + \partial). \] This shows that |$-w+q^2+q^{-2} = \partial$|⁠, as desired. ■ It is clear that there is a surjection |$K_q(T') \twoheadrightarrow K_{q,t}(T^2)$| induced topologically by “filling in the puncture with the special strand.” This means that we can consider the loops |$x',y',z'$| as elements of |$K_{q,t}(T^2)$|⁠. Corollary 3.8. There is an algebra isomorphism |$B_{q,t} \to K_{q,t}(T^2)$| induced by the assignments |$x \mapsto x'$|⁠, |$y \mapsto y'$|⁠, and |$z \mapsto z'$|⁠. □ Proof. First we check that the composition |$f\!:B'_q \to K_q(T') \to K_{q,t}(T^2)$| factors through the quotient |$B'_q \to B_{q,t}$|⁠. To do this we need to check that relation (2.10) holds, that is |$f(w) = \left( \frac t q - \frac q t\right)^2 + \left( q + \frac 1 q\right)^2$|⁠. If |$\partial \in K_q(T')$| is the loop around the puncture in |$T'$|⁠, then |$\partial = -w + q^2 + q^{-2} \in K_q(T')$| (see the proof of Corollary 3.7). Then the third relation in (3.1) implies |$f(\partial) = -q^2t^{-2}-q^{-2}t^2$|⁠, and the calculation |$\left(t/q-q/t\right)^2+(q+q^{-1})^2 = q^2+q^{-2}+q^2t^{-2}+q^{-2}t^2$| shows the claim. To show that |$f:B_{q,t} \to K_{q,t}(T^2)$| is injective, it suffices to show that the kernel of the map |$K_q(T') \to K_{q,t}(T^2)$| is cyclic and is generated by |$\delta + q^2t^{-2}+q^{-2}t^2$|⁠, where |$\delta$| is the loop in |$T'$| that encircles the puncture. This element is exactly the skein relation on the left of Figure 3, and any time this relation appears in a sum of links, we can slide the relation to the top of the manifold |$T'\times [0,1]$|⁠. In other words, any element |$a$| in the kernel of |$K_q(T') \to K_{q,t}(T^2)$| can be written in the form |$a = a'(\delta + q^2t^{-2}+ q^{-2}t^2)$| for some |$a' \in K_q(T')$|⁠. This proves |$\delta + q^2t^{-2}+ q^{-2}t^2$| generates the kernel, as desired. ■ Corollary 3.9. The algebras |$\mathrm{SH}_{q,t}$| and |$K_{q,t}(T^2)$| are isomorphic. □ Proof. Compose the isomorphism of Corollary 3.8 and Theorem 2.27. ■ 3.3 The deformation for 3-manifolds We now define a deformed skein module |$\bar K_{q,t}(M,f)$| for a 3-manifold |$M$| with a boundary component |$F$|⁠, a connected surface. The vector space |$\bar K_{q,t}(M, f)$| is actually a bimodule—it is a left module over |$K_{q,t}(F)$| and a right module over |$K_{q,t}(T^2)$|⁠. The definition of |$\bar K_{q,t}(M, f)$| depends on some additional data (the map |$f$|⁠, which is described below), but if |$M$| is a knot complement |$S^3 \setminus K$|⁠, then we describe a canonical choice for this data. For knot complements we also define a canonical quotient |$K_{q,t}(S^3\setminus K)$| of |$\bar K_{q,t}(S^3\setminus K)$|⁠. This quotient destroys the right module structure, so |$K_{q,t}(S^3\setminus K)$| is just a left module over |$K_{q,t}(T^2)$|⁠. When |$K$| is the unknot this module has the “right size” and can be viewed as a deformation of |$K_q(S^3 \setminus K)$|⁠. However, it is not clear if this is true for other knots. 3.3.1 General 3-manifolds Let |$G$| be the “tadpole” graph depicted in Figure 5, and let |$f:G \to M$| be an embedding with |$f(v_1) = p \in F \subset \partial M$|⁠. (Here |$v_1$| is the vertex of |$G$| that is not on the loop.) As before, the definition does not depend on the choice of |$p \in F$|⁠, but it does depend on the choice of |$f$|⁠. Fig. 5. View largeDownload slide The “tadpole” graph. Fig. 5. View largeDownload slide The “tadpole” graph. Definition 3.10. A framed link compatible with |$\boldsymbol{f}$| is an isotopy class of embeddings |$G \sqcup (\sqcup_n S^1\times[0,1])\hookrightarrow M$| containing a representative such that the embedding of |$G$| is given by the map |$f$|⁠. (The isotopies we consider are those which fix the boundary of |$M$|⁠.) The modified Kauffman bracket skein module|$\boldsymbol{\bar K_{q,t}(M,f)}$| is the vector space of framed links compatible with |$f$| modulo the relations in Equation (3.1), where the third relation in (3.1) is only applied to the arc of the graph |$G$|⁠, and not the loop of |$G$|⁠. □ Remark 3.11. Definition 3.10 could be rephrased as follows. Let |$M' := M \setminus (\textrm{nbhd of } f(G))$|⁠. Then |$\bar K_{q,t}(M)$| is the quotient of the skein module |$K_q(M')$| by the relation that says: if |$L$| is a loop in |$K_q(M')$| that is parallel to the loop on the boundary of |$M'$| that encircles the arc in |$G$|⁠, then |$L = -q^2t^2 - q^{-2}t^{-2}$|⁠. This rephrasing shows that the definition of |$\bar K_{q,t}(M,f)$| is invariant under isotopies of the map |$f$|⁠. □ Lemma 3.12. The space |$\bar K_{q,t}(M, f)$| is a left module over |$K_{q,t}(\partial M)$|⁠. □ Proof. To see this, we note that there is a natural inclusion of the punctured surface |$\partial M \setminus p$| into |$M$|⁠, which means that |$\bar K_{q,t}(M,f)$| is a (left) module over |$K_q( (\partial M) \setminus p)$|⁠. By Remark 3.3, |$K_{q,t}(\partial M)$| is a quotient by |$K_q((\partial M)\setminus p)$| by the relation |$L_p = -q^2t^{-2}-q^{-2}t^2$|⁠. By definition, this relation holds in |$\bar K_{q,t}(M,f)$|⁠, so the action of |$K_q((\partial M)\setminus p)$| factors through to an action of |$K_{q,t}(\partial M)$| on |$\bar K_{q,t}(M,f)$|⁠. ■ If we thicken the loop in the graph |$G$|⁠, then the boundary |$B$| of this thickened loop can be identified with the punctured torus |$T^2 \setminus p$|⁠. Therefore, |$\bar K_{q,t}(M, f)$| is also a right module over |$K_{q,t}(T^2)$|⁠. However, in general this right module structure is not canonical—it depends on the identification of |$T^2 \setminus p$| with the boundary |$B$| (and also on the map |$f$|⁠). 3.3.2 Knot complements Let |$K \subset S^3$| be a knot. In this section we describe a canonical choice of a map |$f:G \to S^3 \setminus K$| (see Figure 6). We also describe a quotient |$K_{q,t}(K)$| of |$\bar K_{q,t}(S^3\setminus K, f)$| that is closely related to the classical skein module |$K_q(S^3\setminus K)$|⁠. In the next section we compute this module when |$K$| is the unknot. Fig. 6. View largeDownload slide The embedding of the graph |$G$| for a knot complement. Fig. 6. View largeDownload slide The embedding of the graph |$G$| for a knot complement. If |$K$| is a knot in |$S^3$|⁠, then there is a unique (up to isotopy) choice of a longitude and meridian in the boundary of a tubular neighborhood of |$K$|⁠. There is therefore a unique (up to isotopy) choice of curve |$C$| in |$S^3\setminus K$| that is parallel to the meridian |$m$|⁠. More precisely, |$C$| is a boundary component of an annulus whose other boundary component is the meridian of |$K$|⁠. We pick a simple arc |$a$| in this annulus connecting |$C$| to the point |$p \in \partial (S^3\setminus K)$|⁠, and we define |$f:G \to S^3\setminus K$| to be the map given by the union of the embeddings of |$C$| and |$a$|⁠. (See Figure 6.) Definition 3.13. If |$K \subset S^3$| is a knot, then |$\bar K_{q,t}(K) := \bar K_{q,t}(S^3\setminus K, f)$|⁠, where the map |$f:G \to S^3\setminus K$| is the map described technically in the previous paragraph (and graphically in Figure 6). □ Remark 3.14. There are many choices of an arc in an annulus connecting the two boundary components, but the definition does not depend on this choice because all such choices are isotopic (via an isotopy that does not fix the embedding of |$G$|⁠). Such isotopies are explicitly allowed in the definition of a “framed link compatible with |$f$|” (see the first sentence of Definition 3.10 and the last sentence of Remark 3.11). □ As was mentioned in the previous section, the vector space |$\bar K_{q,t}(K)$| is a bimodule over |$K_{q,t}(S^1 \times S^1)$|⁠. In general, the right module structure is non-canonical, but with our specific choice of |$f:G \to S^3 \setminus K$|⁠, there is a canonical choice for the right module structure. In more detail, we let |$N_G$| be a closed tubular neighborhood of the loop in the graph |$G$|⁠, and we let |$T_p$| be the boundary of |$N_G$| (which is a punctured torus, where the puncture corresponds to the arc in the graph |$G$|⁠). Up to isotopy there is a unique loop |$m_G$| in |$T_p$| that is contractible in |$N_G$|⁠. By construction, there is a loop |$l_G$| in |$T$| that is isotopic to the meridian of |$K$|⁠, and up to isotopy this loop is unique. (The loops |$m_G$| and |$l_G$| are the meridian and longitude of |$G$| if |$G$| is viewed as the unknot inside of |$S^3$|⁠.) The choice of the (oriented) loops |$m_G$| and |$l_G$| gives an identification of |$S^1 \times S^1$| with |$T$|⁠, and up to isotopy this identification is uniquely determined by the requirement that the first and second factors of |$S^1\times S^1$| are sent to |$m_G$| and |$l_G$|⁠, respectively. We recall that for generic |$q$|⁠, the algebra |$\mathrm{SH}_{q,t}$| is generated by elements |$x,y \in \mathrm{SH}_{q,t}$|⁠, with |$x = (X+X^{-1})\boldsymbol{\mathrm{e}}$| and |$y = (Y+Y^{-1})\boldsymbol{\mathrm{e}}$|⁠. Under the identifications in the previous paragraph, the elements |$x$| and |$y$| act on |$\bar K_{q,t}(K)$| on the left via |$m$| and |$l$|⁠, respectively. The action of |$x$| and |$y$| on the right is given by |$l_G$| and |$m_G$|⁠, respectively. (The meridian |$m$| of |$K$| is isotopic to the longitude |$l_G$| of |$G$|⁠, so the element |$x \in \mathrm{SH}_{q,t}$| acts on the empty link |$1_L \in \bar K_{q,t}(K)$| symmetrically on the left and on the right.) Let |$z_G \in K_{q,t}(T^2)$| be the loop in the boundary of |$G$| that has the coefficient |$q$| in the resolution of the product |$l_Gm_G$|⁠, so that |$z \in \mathrm{SH}_{q,t}$| acts on |$\bar K_{q,t}(K)$| on the right by |$z_G$|⁠. Definition 3.15. Let |$\boldsymbol{\mathrm{e}} P^-$| be the symmetric sign representation, and define the left |$\mathrm{SH}_{q,t}$|-module \[ K_{q,t}(K) := \bar K_{q,t}(K) \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^-. \] □ The module |$K_{q,t}(K)$| can equivalently be defined as the quotient of |$\bar K_{q,t}(K)$| by the left submodule generated by |$L(m_G + tq^{-2}+t^{-1}q^2)$| and |$L(z_G + tq^{-3}l_G)$|⁠, for all |$L \in \bar K_{q,t}(K)$|⁠. Topologically, this definition can be viewed as the |$t$|-analogue of “filling in the graph |$G$|⁠.” More precisely, we have the following lemma. Proposition 3.16. If |$t=1$|⁠, there is a natural surjection |$K_{q,t=1}(K) \twoheadrightarrow K_q(K)$|⁠. □ Proof. There are natural surjections from |$K_q(S^3 \setminus (K\cup G))$| onto |$K_{q,t}(K)$| and |$K_q(K)$|⁠. When |$t=1$|⁠, all relations imposed in the definition of |$K_{q,t=1}(K)$| are also in the kernel of the map |$K_q(S^3\setminus (K \cup G)) \to K_q(K)$|⁠. (This uses the fact that the specialization |$\boldsymbol{\mathrm{e}} P^-_{q,t=1}$| is isomorphic to the skein module of the solid torus.) Therefore this map induces a map |$K_{q,t=1}(K) \to K_q(K)$|⁠. ■ Remark 3.17. The relations defining |$K_{q,t}(K)$| can be viewed as skein relations in some sense, but an important difference is that they are not local. For example, any loop isotopic to the |$(1,1)$| curve in the torus |$T_p$| that bounds the loop in |$G$| is declared to be equal to a loop isotopic to |$l_G$| (which is isotopic to the meridian of |$K$|⁠), and this is not a local relation. This seems to make calculations involving |$K_{q,t}(K)$| more difficult. □ 3.4 The unknot In this section, we compute the |$\mathrm{SH}_{q,t}$|-module |$K_{q,t}(K)$|⁠, where |$K \subset S^3$| is the unknot. We first compute the bimodule structure of |$\bar K_{q,t}(K)$|⁠. The key observation is the following lemma. Lemma 3.18. Let |$U \subset S^3$| be an open |$\epsilon$|-neighborhood of the union of the unknot |$K$| and the graph |$G$|⁠. Then the complement |$S^3 \setminus U$| is diffeomorphic to |$(T^2 \setminus p) \times [0,1]$|⁠. Under this identification, the meridian |$m$| of |$K$| is isotopic to the longitude |$l_G$| of |$G$|⁠, and the longitude |$l$| of |$K$| is isotopic to the meridian |$m_G$| of |$G$|⁠. □ Proof. If we write |$U_a$| for |$U$| with the arc joining |$K$| and the loop in |$G$| removed, then |$U_a$| is the Hopf link, so |$S^3\setminus U_a$| is the thickened torus |$T^2 \times [0,1]$|⁠. Since the arc |$a$| can be embedded in a vertical disc in |$T^2 \times [0,1]$|⁠, it is isotopic to a vertical arc. Therefore, |$S^3 \setminus U$| is diffeomorphic to |$(T^2\setminus p)\times [0,1]$|⁠. ■ Corollary 3.19. We have an isomorphism |$\bar K_{q,t}(K) \cong \mathrm{SH}_{q,t}$| of |$\mathrm{SH}_{q,t}$|-bimodules. □ Proof. Let |$M' = (S^3 \setminus K) \setminus (\textrm{nbhd of } G)$|⁠. By Lemma 3.18, there is a diffeomorphism |$\varphi: M' \to (T^2\setminus p)\times [0,1]$|⁠. By Remark 3.11, |$\bar K_{q,t}(K)$| is the quotient of |$K_q(M')$| by the relation that says: if |$L_p$| is a loop around the arc |$A$| of the graph |$G$|⁠, then |$L_p = -q^2t^{-2}-q^{-2}t^2$|⁠. Since this is exactly the relation defining |$K_{q,t}(T^2)$|⁠, this shows that |$\varphi$| induces an isomorphism of vector spaces, which implies |$\bar K_{q,t}(K) $| is isomorphic to |$\mathrm{SH}_{q,t}$| as vector spaces. The fact that |$\varphi$| respects the bimodule structure follows from the third sentence of Lemma 3.18 and the discussion following Remark 3.14. ■ Theorem 3.20. The left module |$K_{q,t}(K)$| is isomorphic to |$\boldsymbol{\mathrm{e}} P^-$|⁠. □ Proof. By Definition 3.15, |$K_{q,t}(K) = \bar K_{q,t} \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^-$|⁠. Then Corollary 3.19 shows that |$K_{q,t}(K) = \mathrm{SH}_{q,t}\otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^-$|⁠, which completes the claim. ■ We write |$\phi(P^-)$| for the twist of |$P^-$| by the anti-automorphism |$\phi$| defined in (2.20). Corollary 3.21. If |$K$| is the unknot, the vector space |$ \phi(P^-)\boldsymbol{\mathrm{e}} \otimes_{\mathrm{SH}_{q,t}} \bar K_{q,t}(K) \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^-$| is one-dimensional. □ Proof. By Corollary 3.19, there is a bimodule isomorphism |$\bar K_{q,t}(K) \cong \mathrm{SH}_{q,t}$|⁠. By Lemma 2.37, the space of linear functions |$\phi(P^-)\boldsymbol{\mathrm{e}} \otimes_{\mathrm{SH}_{q,t}} \boldsymbol{\mathrm{e}} P^- \to {\mathbb{C}}$| is one-dimensional, which proves the claim. ■ Remark 3.22. It is natural to ask whether this corollary can be generalized to non-trivial knots. This seems like a subtle question in general—in particular, the proof in the case of the unknot relies on the existence and (essential) uniqueness of Cherednik’s pairing (see Lemma 2.37), which is a nontrivial fact. □ 4 A Topological Interpretation of Cherednik’s Construction In Section 2.1.5, we described how the colored Jones polynomials of a cable of a knot |$K$| can be computed from the skein module |$K_q(S^3\setminus K)$|⁠. In particular, the colored Jones polynomials of torus knots can be computed from the skein module of the unknot. In this section, we use the topological construction of the previous sections to define polynomials |$J_{n,r,s}(q,t)$| that satisfy \begin{equation}\label{toppolys} J_{n,r,s}(q,t=1) = J_{n}(K_{r,s};q), \end{equation} (4.1) where |$K_{r,s}$| is the |$(r,s)$|-torus knot. (Technically, |$J_{n,r,s}$| is a rational function of |$q$| and |$t$|⁠, but this is because the Macdonald polynomials are rational functions of |$q$| and |$t$|⁠). Cherednik [14] used the double affine Hecke algebra of type |${\mathfrak{g}}$| to construct two-variable polynomials that specialize to the colored Jones polynomials of type |${\mathfrak{g}}$|⁠. We recall his construction for |${\mathfrak{g}} = {\mathfrak{sl}}_2$| and then show his polynomials are equal to the polynomials |$J_{n,r,s}(q,t)$| (up to a renormalization). Remark 4.1. It turns out that Cherednik’s parameter |$t$| is slightly different than ours. To try to eliminate confusion, in this section we will write |${t_c} \in {\mathbb{C}}^*$| for Cherednik’s parameter and |$t \in {\mathbb{C}}^*$| for ours. In general we will use a subscript |$c$| to indicate objects which depend on Cherednik’s parameter. □ 4.1 Cherednik’s construction We first recall Cherednik’s construction from [14]. Let |$P^+_c = {\mathbb{C}}[X^{\pm 1}]$| be the polynomial representation of |${\mathrm{H}}_{q,{t_c}}$|⁠, and define the evaluation map \begin{equation*} \epsilon_c: P^+_c \to {\mathbb{C}},\quad \epsilon_c(f(X)) = f(t_c). \end{equation*} In the notation of Section 2.35, we can write this evaluation map as |$\epsilon(-) = \langle 1, -\rangle_c$|⁠, where the pairing on the right is defined in Corollary 2.35. If we identify |$\boldsymbol{\mathrm{e}} P^+_c \cong {\mathbb{C}}[x] = {\mathbb{C}}[X+X^{-1}]$| and restrict the evaluation map, for |$g(x) \in {\mathbb{C}}[x]$| we obtain \begin{equation}\label{cheredniksevaluation} \epsilon_c: \boldsymbol{\mathrm{e}} P^+_c \to {\mathbb{C}},\quad \epsilon_c(g(x)) = \epsilon_c(g(X+X^{-1})) = g(t_c+t^{-1}_c). \end{equation} (4.2) We recall Cherednik’s construction of an |${\rm{SL}}_2({\mathbb{Z}})$| action on |${\mathrm{H}}_{q,{t_c}}$| via the following formulas: \begin{align}\label{cherednikssl2zaction} \left[\begin{array}{cc}1&1\\0&1\end{array}\right] &\mapsto \tau^+, &\quad \tau^+(X) &= X, &\quad \tau^+(T) &=T, &\quad \tau^+(Y) &= q^{-1}XY\\ \left[\begin{array}{cc}1&0\\1&1\end{array}\right] &\mapsto \tau^-, &\quad \tau^-(X) &= qYX, &\quad \tau^-(T) &=T, &\quad \tau^-(Y) &= Y.\notag \end{align} (4.3) Let |$\gamma_{r,s} \in {\rm{SL}}_2({\mathbb{Z}})$| be a matrix satisfying |$\gamma_{r,s}(0,1) = (r,s)$|⁠, and let |$p_n \in {\mathbb{C}}[X+X^{-1}] \subset {\mathbb{C}}[X^{\pm 1}]$| be the Macdonald polynomial of Theorem 2.33. Cherednik then gives the following definition: Definition 4.2 ([14]). The nonreduced DAHA-Jones invariant is defined by \begin{equation}\label{cherednikpoly} P_{n, r,s}(q,t_c) := (-q)^{rs(n^2-1)}\epsilon_c(\gamma_{r,s}(p_{n-1}(Y+Y^{-1}))\cdot 1). \end{equation} (4.4) □ These polynomials do not depend on the choice of |$\gamma_{r,s}$| (see Lemma 5.6). Remark 4.3. The polynomials above are defined in [14, (2.18)], but we have changed the normalization by a power of |$-q$|⁠. His |$b$| is our |$n-1$|⁠, so |$b(b+2) = n^2-1$|⁠. Also, Cherednik’s |$q^{1/2}$| is our |$q^2$|⁠, and his |$t^{1/2}$| is our |$t$|⁠. Our Jones polynomials are normalized so that the |$n^\mathrm{th}$| colored Jones polynomial of the unknot is |$(-1)^{n-1}(q^{2n}-q^{-2n}) / (q^2-q^{-2})$|⁠. □ 4.2 The topological construction We now give a topological interpretation of these polynomials (after establishing some notation). Let |$K$| be the unknot. Recall from (2.21) that as a |${\mathbb{C}}[x]$|-module we have |$K_{q,t}(K) = {\mathbb{C}}[x]\delta_t$|⁠, and that by Theorem 3.20 the action of |$\mathrm{SH}_{q,t}$| is determined by \begin{equation}\label{unknotaction} y\cdot \delta_t = -(tq^{-2}+t^{-1}q^2)\delta_t,\quad z\cdot \delta_t = -q^{-3}t^{-1}x\delta_t. \end{equation} (4.5) We first compare the module |$K_{q,t}(K)$| to the classical skein module |$K_q(K)$|⁠. Lemma 4.4. When |$K$| is the unknot, the |$A_q^{{\mathbb{Z}}_2}$|-modules |$K_q(K)$| and |$K_{q,t=1}(K)$| are isomorphic. □ Proof. The formulas in (4.5) follow directly from Theorem 3.20. As a |${\mathbb{C}}[x]$|-module, the classical skein module |$K_q(K)$| is freely generated by the empty link. If the formulas (4.5) are specialized to |$t=1$|⁠, then the action of |$y$| and |$z$| on |$\delta_t$| agrees with the action of |$y$| and |$z$| on the empty link. Then Lemma 2.29 shows that the |${\mathbb{C}}[x]$|-module isomorphism sending |$\delta_{t=1}$| to the empty link is an isomorphism of |$A_q^{{\mathbb{Z}}_2}$|-modules. ■ We now need the analogue of the topological pairing in (2.2) which is provided by Lemma 2.37. We recall that this pairing is defined by \begin{equation} \langle -,-\rangle_t: \phi(P^-)\boldsymbol{\mathrm{e}} \otimes_{\mathrm{SH}_{q,t}} K_{q,t}(K) \to {\mathbb{C}},\quad \langle \delta_t, \varnothing\rangle = 1. \end{equation} (4.6) (Recall that |$\phi$| is the anti-automorphism defined in (2.20), and |$\delta_t = 1\cdot (Y\boldsymbol{\mathrm{e}})$| is the generator of |$\phi(P^-)\boldsymbol{\mathrm{e}}$|⁠, and |$\varnothing$| is the empty link in |$K_{q,t}(K)$|⁠.) Lemma 4.5. When |$t=1$|⁠, the pairing |$\langle -,-\rangle_{t=1}$| is the same as the topological pairing from (2.2). □ Proof. The topological pairing is uniquely determined by the requirement |$\langle \varnothing, \varnothing\rangle = 1$|⁠, and the algebraic pairing is uniquely determined by the requirement |$\langle \delta_t,\delta_t\rangle_t = 1$|⁠. Furthermore, the isomorphism |$K_{q,t}(K) \to \boldsymbol{\mathrm{e}} P^-$| satisfies |$\varnothing \mapsto \delta_t$|⁠, which completes the proof. ■ To simplify notation, we will view this pairing as a functional on |$K_{q,t}(K)$|⁠: \begin{equation}\label{ourevaluation} \epsilon: K_{q,t}(K) \to {\mathbb{C}},\quad\quad \epsilon(u) = \langle \delta_t, u\rangle_t. \end{equation} (4.7) Lemma 4.6. We have the equality |$\epsilon(f(x)) = f(-tq^{-2}-t^{-1}q^2)$|⁠. □ Proof. This follows from the identity |$y\cdot \delta_t = -(tq^{-2}+t^{-1}q^2)\delta_t$| and the fact that |$\phi(x) = y$|⁠. ■ Definition 4.7. Let |$p^-_n(x) \in {\mathbb{C}}[x]$| be the sign Macdonald polynomials of Definition 2.38, and let |$\gamma_{r,s} \in {\rm{SL}}_2({\mathbb{Z}})$| satisfy |$\gamma_{r,s}(0,1) = (r,s)$|⁠. We then define \begin{equation} J_{n,r,s}(q,t) := (-q)^{rs(n^2-1)}\epsilon(\gamma_{r,s}(p^-_{n-1}(y))\cdot \varnothing) \end{equation} (4.8) □ These polynomials do not depend on the choice of |$\gamma_{r,s}$| by Lemma 5.6. Remark 4.8. Lemma 4.5 shows that our pairing can be interpreted as the |$t$|-analogue of “gluing the knot |$K$| into |$S^3\setminus K$| to obtain |$S^3$|⁠.” Under this interpretation, the polynomials |$J_{n,r,s}(q,t)$| can be interpreted (roughly) as the evaluations of parallels of the |$r,s$| torus knot embedded in |$S^3\setminus (\mathrm{unknot } \cup G)$|⁠. □ 4.3 The comparison In this section, we relate Cherednik’s polynomials |$P_{n,r,s}(q,t_c)$| to the polynomials |$J_{n,r,s}(q,t)$| that were constructed in the previous section using the cabling formula. We also relate these polynomials to the colored Jones polynomials |$J_n(K_{r,s}; q)$| of the |$(r,s)$| torus knot. Theorem 4.9. We have the following equalities \begin{align*} P_{n,r,s}(q,t_c=-q^2t^{-1}) &= J_{n,r,s}(q,t) \\ J_n(K_{r,s}; q) &= P_{n,r,s}(q,t_c=-q^2) = J_{n,r,s}(q, t=1). \end{align*} □ Remark 4.10. The equality |$J_n(K_{r,s}; q) = P_{n,r,s}(q, t=-q^2)$| in Theorem 4.9 was proved in [14, Theorem 2.8] by direct computation without relying on skein modules or the cabling formula. We also remark that the key fact used to prove the first equality is an isomorphism of algebras |$\mathrm{SH}_{q,t=-q^2} \cong \mathrm{SH}_{q,t=1}$|⁠, and that the standard polynomial representation at |$t=-q^2$|⁠, when transferred along this isomorphism, becomes the sign representation. Then the second and third equalities are proved using the cabling formula, along with an identification of |$\mathrm{SH}_{q,t=1} = A_q^{{\mathbb{Z}}_2} \cong K_q(T^2)$|⁠. □ We begin with some lemmas which will be used in the proof. First, we recall the algebra |$B'_q$|⁠, which is generated by elements |$x,y,z$| subject to the relations \[ [x,y]_q = (q^2-q^{-2})z,\quad [z,x]_q = (q^2-q^{-2})y,\quad [y,z]_q = (q^2-q^{-2})x. \] From Theorem 2.27, there is a surjection |$f:B'_q \to \mathrm{SH}_{q,t}$| given by \begin{equation}\label{mapfrombq} f(x) = (X+X^{-1})\boldsymbol{\mathrm{e}},\quad f(y) = (Y+Y^{-1})\boldsymbol{\mathrm{e}},\quad f(z) = q^{-1}(XYT^{-2}+X^{-1}Y^{-1})\boldsymbol{\mathrm{e}}. \end{equation} (4.9) We first relate the |${\rm{SL}}_2({\mathbb{Z}})$| actions that are used in constructing both polynomials. The mapping class group |${\rm{SL}}_2({\mathbb{Z}})$| of |$T^2 \setminus p$| acts on |$K_q(T^2\setminus p)$|⁠, and we can transport this action to |$B'_q$| using the isomorphism |$B'_q = K_q(T^2\setminus p)$| of Theorem 3.5. We define two automorphisms |$\tau_\pm :B'_q \to B'_q$|⁠: \begin{align}\label{sl2actiononB} \left[\begin{array}{cc}1&1\\0&1\end{array}\right] &\mapsto \tau^+, &\quad \tau^+(x) &= x, &\quad \tau^+(y) &=z, &\quad \tau^+(z) &= q^{-1}xz-q^{-2}y\\ \left[\begin{array}{cc}1&0\\1&1\end{array}\right] &\mapsto \tau^-, &\quad \tau^-(x) &= z, &\quad \tau^-(y) &=y, &\quad \tau^-(z) &= q^{-1}zy - q^{-2}x.\notag \end{align} (4.10) Lemma 4.11. The mapping class group |${\rm{SL}}_2({\mathbb{Z}})$| acts on |$B'_q$| via formula (4.10). The induced action of |${\rm{SL}}_2({\mathbb{Z}})$| on |$\mathrm{SH}_{q,t}$| agrees with Cherednik’s |${\rm{SL}}_2({\mathbb{Z}})$| action defined in (4.3). □ Proof. These are all straightforward computations. ■ We next relate the |$\mathrm{SH}_{q,t_c}$|-module |$\boldsymbol{\mathrm{e}}_c P_c^+$| used in Cherednik’s construction to the module |$K_{q,t}(K)\cong \boldsymbol{\mathrm{e}}_t P^-$| used in the topological construction. We consider both these modules as modules over |$B'_q$| using the surjection |$f$| in (4.9). We will write |$\boldsymbol{\mathrm{e}}_c \in {\mathrm{H}}_{q,t_c}$| and |$\boldsymbol{\mathrm{e}}_t \in {\mathrm{H}}_{q,t}$| for the idempotent |$(T+t^{-1})/(t+t^{-1})$|⁠. With this notation, we recall that we have the following equalities of |${\mathbb{C}}[x]$|-modules: \begin{equation}\label{isoppluspminus} \boldsymbol{\mathrm{e}}_c P_c^+ = {\mathbb{C}}[x] 1_c \subset {\mathbb{C}}[X^{\pm 1}],\quad \boldsymbol{\mathrm{e}}_t P^- = {\mathbb{C}}[x]\delta_t \subset {\mathbb{C}}[X^{\pm 1}]. \end{equation} (4.11) The modules |$\boldsymbol{\mathrm{e}}_c P_c^+$| and |$\boldsymbol{\mathrm{e}}_t P^-$| are isomorphic as |${\mathbb{C}}[x]$|-modules via the isomorphism \[ \varphi:\boldsymbol{\mathrm{e}}_c P_c^+ \to \boldsymbol{\mathrm{e}}_t P^-, \quad \quad \varphi(1_c) = \delta_t. \] Lemma 4.12. If |$t_c = -q^2t^{-1}$|⁠, then the map |$g: \boldsymbol{\mathrm{e}}_{c} P_c^+ \to \boldsymbol{\mathrm{e}}_t P^-$| is an isomorphism of |$B'_q$|-modules. This map satisfies |$\varphi(p_n(x)) = p^-_n(x)\delta_t$|⁠. □ Proof. A short computation shows that \[ y\cdot 1_c = (t_c+t_c^{-1})1_c,\quad z\cdot 1_c = q^{-1}t^{-1}x1_c. \] Another short computation shows that \[ y\cdot \delta_t = -(tq^{-2}+t^{-1}q^2)\delta_t,\quad z\cdot \delta_t = -tq^{-3} x\delta_t. \] If |$t_c = -q^2t^{-1}$|⁠, then these formulas agree, and Lemma 2.29 gives the first claim. The second claim follows from the fact that the Macdonald polynomials |$p_n(x)$| and |$p_n^-(x)$| are an eigenbasis for the operator |$y$| for the modules |$\boldsymbol{\mathrm{e}}_c P_c^+$| and |$\boldsymbol{\mathrm{e}}_t P^-$|⁠, respectively. ■ Finally, we relate the evaluation maps (4.2) and (4.7) that are used to define the polynomials |$P_{n,r,s}$| and |$J_{n,r,s}$|⁠. Lemma 4.13. If |$t_c = -q^2t^{-1}$|⁠, then for all |$h(x) \in \boldsymbol{\mathrm{e}}_c P^+_c$| we have \[ \epsilon_c(h(x)) = \epsilon(\varphi(h(x)). \] □ Proof. From (4.2), we have |$\epsilon_c(h(x)) = h(t_c+t_c^{-1})$|⁠. When |$t_c = -q^2t^{-1}$| this agree with the equality |$\epsilon(h(x)) = h(-tq^{-2}-t^{-1}q^2)$| from (4.7). ■ We have now collected the facts needed to prove Theorem 4.9. We first recall the equalities claimed in the theorem: \begin{align*} P_{n,r,s}(q,t_c = -q^2t^{-1}) &= J_{n,r,s}(q,t) \\ J_n(K_{r,s}; q) &= P_{n,r,s}(q,t_c=-q^2) = J_{n,r,s}(q, t=1). \end{align*} Proof. (of Theorem 4.9) We recall that the polynomials |$P_{n,r,s}(q,t_c)$| and |$J_{n,r,s}(q,t)$| are defined as follows: \begin{align*} P_{n,r,s}(q,t_c) := (-q)^{rs(n^2-1)}\epsilon_c(\gamma_{r,s}(p_{n-1}(y))\cdot 1_c)\\ J_{n,r,s}(q,t) := (-q)^{rs(n^2-1)}\epsilon(\gamma_{r,s}(p^-_{n-1}(y))\cdot \varnothing)). \end{align*} The empty link is denoted |$\varnothing$|⁠, and under the identification |$K_{q,t}(K) = \boldsymbol{\mathrm{e}}_t P^-$| we have |$\varnothing \mapsto \delta_t$|⁠. By Lemma 4.12, we see that |$\varphi(y\cdot1_c) = y\cdot \varphi(1_c) = y\cdot \delta_t$|⁠. Then Lemma 4.11 shows that |$\varphi(\gamma_{r,s}(p_n(y))\cdot 1_c) = \gamma_{r,s}(p_n(y))\cdot \delta_t$|⁠. Finally, Lemma 4.13 completes the proof of the first equality. To complete the proof of the theorem, we first note that when |$t=1$|⁠, the module |$K_{q,t=1}(K)$| and the pairing |$\langle -,-\rangle_t$| are the same as the classical skein module |$K_q(K)$| and the classical topological pairing. (These claims are Lemma 4.4 and 4.5, respectively.) The second equality in the statement of the theorem then follows from the cabling formula in Corollary 2.16. ■ 5 Iterated Cablings of the Unknot The key facts that allowed the comparison of Cherednik’s polynomials to colored Jones polynomials of torus knots are that the colored Jones polynomials satisfy a cabling formula and that torus knots are cables of the unknot. However, the cabling formula applies to all knots, and in particular can be used to write colored Jones polynomials of iterated cables of the unknot in terms of colored Jones polynomials of the unknot. In this section, we use this observation to extend Cherednik’s construction. More precisely, let |${\mathbf{r}} = (r_1,\ldots,r_m)$| and |${\mathbf{s}} = (s_1,\ldots, s_m)$| with |$r_i,s_i \in {\mathbb{Z}}$| relatively prime, and write |${\mathbf{r}}_{k} = (r_1,\ldots,r_k)$| and similarly for |${\mathbf{s}}_k$|⁠. Definition 5.1. Let |$K^{\mathrm{top}}({\mathbf{r}}_1,{\mathbf{s}}_1)$| be the 0-framed |$(r_1,s_1)$| torus knot, and let |$K^{\mathrm{top}}({\mathbf{r}}_m,{\mathbf{s}}_m)$| be the |$(r_m,s_m)$| topological cable of the knot |$K^{\mathrm{top}}({\mathbf{r}}_{m-1},{\mathbf{s}}_{m-1})$|⁠. (See Definition 2.13—in particular, each |$K^{\mathrm{top}}({\mathbf{r}}_{k},{\mathbf{s}}_k)$| is 0-framed.) □ Below we will define two-variable polynomials |$J_n({\mathbf{r}},{\mathbf{s}}; q,t) \in {\mathbb{C}}[q^{\pm 1},t^{\pm 1}]$|⁠, and we will show that they specialize to the colored Jones polynomials of the knot |$K^{\mathrm{top}}({\mathbf{r}},{\mathbf{s}})$|⁠: \begin{equation} J_n({\mathbf{r}},{\mathbf{s}}; q,t=-q^2) = J_n(K^{\mathrm{top}}({\mathbf{r}},{\mathbf{s}}); q). \end{equation} (5.1) When |$m = 1$|⁠, this construction reproduces Cherednik’s construction in [14]. Remark 5.2. As we learned in the last section, to produce Jones polynomials using the DAHA we can either use the sign representation and specialize to |$t=1$|⁠, or we can use the standard polynomial representation and specialize to |$t=-q^2$|⁠. In this section, we will make the latter choice (since for higher rank DAHAs this specialization is more natural). □ Before proceeding, we give an extension of Corollary 2.16 to iterated cables of the 0-framed unknot. Remark 5.3. In Definition 2.14, the identification of the skein modules of the neighborhoods |$N_K$| and |$N_{r,s}$| of a knot |$K$| and its cable |$K_{r,s}$| are the same since they are both 0-framed. We may therefore view the map |$\Gamma_{r,s}^{\mathrm{top}}$| of equation (2.4) as a map |$\Gamma_{r,s}^{\mathrm{top}}: {\mathbb{C}}[y] \to {\mathbb{C}}[y]$| by identifying |$u = y$|⁠. □ Corollary 5.4. Given sequences |${\mathbf{r}}$|⁠, |${\mathbf{s}}$| as above, let |$\Gamma^{\mathrm{top}}_{{\mathbf{r}},{\mathbf{s}}} = \Gamma^{\mathrm{top}}_{r_1,s_1} \circ \cdots \circ \Gamma^{\mathrm{top}}_{r_m,s_m}: {\mathbb{C}}[y] \to {\mathbb{C}}[y]$|⁠, where |$\Gamma^{\mathrm{top}}_{r,s}$| is defined in equation (2.4). Then we have the equality \[ J_n(K^{\mathrm{top}}({\mathbf{r}},{\mathbf{s}})) = \langle \varnothing \cdot \Gamma^{\mathrm{top}}_{{\mathbf{r}},{\mathbf{s}}}(S_{n-1}(y)),\varnothing\rangle_{unknot}. \] □ Proof. This follows from the proof of Corollary 2.16 combined with a simple induction. ■ 5.1 Two-variable polynomials for iterated cables Let |$\phi:{\mathrm{H}}_{q,t} \to {\mathrm{H}}_{q,t}$| be the anti-automorphism from (2.20). Since |$\phi(T) = T$|⁠, this anti-automorphism restricts to the spherical subalgebra |$\mathrm{SH}_{q,t}$|⁠. In terms of the generators |$x,y,z \in \mathrm{SH}_{q,t}$| from Theorem 2.27, we have \[ \phi(x) = y,\quad \phi(y) = x,\quad \phi(z) = z. \] Let |$P$| be the polynomial representation of |${\mathrm{H}}_{q,t}$| defined in Section 2.2.3 twisted by the anti-automorphism |$\phi$|⁠, and let |$P\boldsymbol{\mathrm{e}}$| be its symmetrization. Explicitly, we may identify |$P\boldsymbol{\mathrm{e}} = {\mathbb{C}}[y]$|⁠, where the action of |$\mathrm{SH}_{q,t}$| on |$P\boldsymbol{\mathrm{e}}$| is determined by the formulas \begin{equation} f(y)\cdot y = f(y)y,\quad 1\cdot x = (t+t^{-1}),\quad 1\cdot z = q^{-1}t^{-1}y. \end{equation} (5.2) Recall that |${\rm{SL}}_2({\mathbb{Z}})$| acts on |$\mathrm{SH}_{q,t}$| via (4.3). Definition 5.5. Given |$r,s \in {\mathbb{Z}}$| relatively prime, let |$\gamma_{r,s} \in {\rm{SL}}_2({\mathbb{Z}})$| be such that \[ \gamma_{r,s}\left(\begin{array}{c} 0\\1\end{array}\right) = \left(\begin{array}{c} r\\s\end{array}\right)\!\!. \] □ Lemma 5.6. The element |$\gamma_{r,s}(y) \in \mathrm{SH}_{q,t}$| depends only on |$r,s$| and not on the choice of |$\gamma_{r,s}$|⁠. □ Proof. The stabilizer of |$(0,1)^T$| in |${\rm{SL}}_2(Z)$| is the subgroup generated by |$\tau^-$| (see formula (4.3)). Since |$\tau^-(Y) = Y$|⁠, we have |$\tau^-(y) = y$|⁠, which proves the claim. ■ We let |$p_n(y) \in P\boldsymbol{\mathrm{e}}$| be the standard Macdonald polynomials defined in Theorem 2.33. To define our polynomials we will need the map |$\iota_{q,t}$| of right |${\mathbb{C}}[Y+Y^{-1}]$|-modules and the evaluation map |$\epsilon_{q,t}:P\boldsymbol{\mathrm{e}} \to {\mathbb{C}}$|⁠: \begin{equation}\label{eq_iotaqt} \iota_{q,t}: P\boldsymbol{\mathrm{e}} \to \mathrm{SH}_{q,t},\quad \quad \iota_{q,t}(f(y)) = f(y), \quad \quad \epsilon_{q,t}(v) = \langle v \mid 1\rangle_{q,t}, \end{equation} (5.3) where the pairing defining |$\epsilon_{q,t}$| is the one from Corollary 2.35. Definition 5.7. Let |$r,s \in {\mathbb{Z}}$| be relatively prime. Define a |${\mathbb{C}}$|-linear map |$\Gamma^{\mathrm{top}}_{r,s;q,t}:P\boldsymbol{\mathrm{e}} \to P\boldsymbol{\mathrm{e}}$| by \[ \Gamma^{\mathrm{top}}_{r,s;q,t}(p_{n-1}(y)) := (-q)^{rs(n^2-1)}1 \cdot \gamma_{r,s}(\iota_{q,t}(p_{n-1}(y))). \] Given sequences |${\mathbf{r}} = (r_1,\cdots,r_m)$| and |${\mathbf{s}} = (s_1,\cdots,s_m)$| with |$r_i,s_i \in {\mathbb{Z}}$| relatively prime, define \[ \Gamma^{\mathrm{top}}_{{\mathbf{r}},{\mathbf{s}};q,t} := \Gamma^{\mathrm{top}}_{r_1,s_1;q,t}\circ \cdots \circ \Gamma^{\mathrm{top}}_{r_m,s_m;q,t},\quad \quad J_n({\mathbf{r}},{\mathbf{s}};q,t) := \epsilon_{q,t}(\Gamma^{\mathrm{top}}_{{\mathbf{r}},{\mathbf{s}};q,t}(p_{n-1}(y))). \] □ Theorem 5.8. We have the equality \[ J_n({\mathbf{r}},{\mathbf{s}};q,t=-q^2) = J_n(K({\mathbf{r}},{\mathbf{s}}); q). \] □ Proof. By Theorem 2.27, the algebra |$\mathrm{SH}_{q,t=1}$| and |$\mathrm{SH}_{q,t=-q^2}$| are isomorphic, and by Lemma 4.11 this isomorphism is |${\rm{SL}}_2({\mathbb{Z}})$|-equivariant. By Lemma 4.12 the right |$K_q(T^2)$|-module |$K_q(N_K)$| is isomorphic to the right |$\mathrm{SH}_{q,t=-q^2}$|-module |$P\boldsymbol{\mathrm{e}}$|⁠. The inclusion maps |$\iota_{q,t=-q^2}$| from (5.3) and |$\iota$| from (2.3) agree, and when |$t=-q^2$| the Macdonald polynomials |$p_n(y)$| specialize to the Chebyshev polynomials |$S_n(y)$|⁠. Therefore, the linear map |$\Gamma_{r,s;q,t=-q^2}^{\mathrm{top}}$| is equal to the linear map |$\Gamma_{r,s}^{\mathrm{top}}$| of (2.4), which implies that |$\Gamma_{{\mathbf{r}},{\mathbf{s}};q,t=-q^2}^{\mathrm{top}} = \Gamma_{{\mathbf{r}},{\mathbf{s}}}^{\mathrm{top}}$|⁠. Finally, |$\epsilon_{q,t=-q^2}(a) = \langle a,\varnothing\rangle_{unknot}$| by Lemmas 4.5 and 4.13. This means that when |$t=-q^2$|⁠, the formula defining |$J_n({\mathbf{r}},{\mathbf{s}};q,t)$| agrees exactly with the cabling formula for |$J_n(K^{\mathrm{top}}({\mathbf{r}},{\mathbf{s}}))$| in Corollary 5.4. This completes the proof of the theorem. ■ 5.2 Examples In this section, we include example calculations of the polynomials constructed in the previous section. We first consider the case of the knot |$Kp := K({\mathbf{r}},{\mathbf{s}})$| where |${\mathbf{r}} = (2,2)$| and |${\mathbf{s}} = (3,5)$|⁠, which is the |$(2,5)$| cable of the |$(2,3)$| knot (i.e., the trefoil). (We choose this example because it is the simplest example in which the ordering of the |$\Gamma_{r_i,s_i}$| matters.) To produce this knot and calculate its (normalized) colored Jones polynomial in Mathematica using the KnotTheory package, one inputs \begin{equation}\label{eq_jones25of23} \begin{array}{l} \textbf{Kp} = \textbf{BR}[4,\{2,1,3,2,2,1,3,2,2,1,3,2,-1\}]\\ \textbf{ColouredJones}\ [\textbf{Kp},1][q^4](q^4-q^{-4})/(q^2-q^{-2})\\ q^{14}+q^{18}+q^{22}+q^{26}-q^{42}-q^{46}-q^{50}+q^{58}. \end{array} \end{equation} (5.4) The output of this is the first image in Figure 7 along with the polynomial displayed in (5.4). Similarly, for the |$(2,-5)$| cable |$Km$| of the trefoil, the following produces the second image of Figure 7: \begin{equation}\label{eq_jones2m5of23} \begin{array}{l} \textbf{Km} = \textbf{BR}[4,\{ 2,1,3,2,2,1,3,2,2,1,3,2, -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1\}]\\ \textbf{ColouredJones}\ [\textbf{Km},1][q^4](q^4-q^{-4})/(q^2-q^{-2})\\ -q^{-30}+q^{-6}+q^{-2}+q^2+q^6+q^{10}-q^{22}-q^{26}-q^{30}+q^{38}. \end{array} \end{equation} (5.5) Fig. 7. View largeDownload slide The |$(2,5)$| and |$(2,-5)$| cables of the |$(2,3)$| knot. Fig. 7. View largeDownload slide The |$(2,5)$| and |$(2,-5)$| cables of the |$(2,3)$| knot. Remark 5.9. In the definition of the |$(r,s)$| cable of a knot |$K$|⁠, one identifies the standard torus |$S^1\times S^1$| with the boundary of a neighborhood |$N_K$| of |$K$|⁠. This identification is done using Lemma 2.1—in other words, the longitude of the boundary torus of |$N_K$| must have |$0$| linking number with the knot |$K$|⁠. This explains why the first image in Figure 7 has one “negative” crossing at the right, instead of five “positive” crossings (since six negative crossings must be added to correct for the framing/linking). Similarly, the second image of Figure 7 has eleven negative crossings instead of five. □ We have also implemented code to compute the polynomials of Definition 5.7 in Mathematica using the NCAlgebra package. Running this code produces the following polynomials: \begin{eqnarray*} (1 - q^4 t^2)J_2(Kp; q,t) &=& q^{32} \left(\!\frac{1}{t}-t^3\!\right)+ q^{44} \left(-\frac{1}{t^{15}}-\frac{1}{t^{13}}\right)+ q^{48} \left(\frac{1}{t^{11}}+\frac{1}{t^9}\right)+ q^{52} \left(\!-\frac{1}{t^{13}}+\frac{1}{t^9}\!\right)\\ &{ }& +q^{56} \left(-\frac{1}{t^{13}}+\frac{2}{t^9}-\frac{1}{t^5}\right)+ q^{60} \left(-\frac{1}{t^{11}}+\frac{1}{t^9}+\frac{1}{t^7}-\frac{1}{t^5}\right)\\ &{ }& +q^{64} \left(\frac{1}{t^9}+\frac{1}{t^7}-\frac{1}{t^5}-\frac{1}{t^3}\right)+ q^{68} \left(-\frac{1}{t^5}+\frac{1}{t}\right) \\ (1 - q^4 t^2)J_2(Km; q,t) &=& q^{-28}\left(\frac{1}{t}-t^3\right) + q^4 \left(-\frac{1}{t^5}-\frac{1}{t^3}\right)+ q^8 \left(\frac{1}{t}+t\right)+ q^{12} \left(-\frac{1}{t^3}+t\right)\\ &{ }&+ q^{16} \left(-\frac{1}{t^3}+2 t-t^5\right)+ q^{20} \left(-\frac{1}{t}+t+t^3-t^5\right)\\ &{ }&+ q^{24} \left(t+t^3-t^5-t^7\right)+ q^{28} \left(-t^5+t^9\right). \end{eqnarray*} For completeness, we also include the following polynomials for the trefoil |$K_{2,3}$|⁠: \begin{eqnarray*} J_2(K_{2,3};q,t) &=& q^{12} \left(\frac{1}{t^5}+\frac{1}{t^3}\right)+q^{16} \left(\frac{1}{t^3}-t\right)\\ (1 - q^4 t^2)J_3(K_{2,3};q,t) &=& q^{24} \left(-\frac{1}{t^{10}}-\frac{1}{t^8}\right)+q^{32} \left(-\frac{1}{t^8}+\frac{1}{t^4}\right)+q^{28} \left(\frac{1}{t^6}+\frac{1}{t^4}\right)\\ & { } & +\,q^{36} \left(-1-\frac{1}{t^8}+\frac{2}{t^4}\right)+ +q^{40} \left(-1-\frac{1}{t^6}+\frac{1}{t^4}+\frac{1}{t^2}\right)\\ &{ }&+\, q^{44} \left(-1+\frac{1}{t^4}+\frac{1}{t^2}-t^2\right)+q^{48} \left(-1+t^4\right). \end{eqnarray*} Funding The work of the author was funded in part by the European Research Council grant no. 637618. Acknowledgements We would like to thank Yuri Berest for extensive explanations and discussions while advising the author’s thesis [30] (which contained the results in the first three subsections of Section 3), and for providing notes on which Section 2.2 was based. We would also like to thank I. Cherednik for helpful comments on an early version of this article and discussions of [15], and G. Masbaum for several conversations, and in particular for explaining the sign in Corollary 2.15. We also thank D. Bar-Natan, E. Gorsky, A. Marshall, G. Muller, A. Oblomkov, M. Pabiniak, S. Shakirov, and D. Thurston for enlightening conversations. The author is grateful to the users of the website MathOverflow who have provided several helpful answers (see, e.g., [4]). Finally, several computations were done using Mathematica, and in particular using the packages KnotTheory and NCAlgebra. Appendix A: Comparing polynomials for iterated cables After the first version of the present article appeared, Cherednik and Danilenko [15] gave a construction of certain polynomials for general |${\mathfrak{g}}$|⁠. For |${\mathfrak{g}} = {\mathfrak{sl}}_2$|⁠, the constructions in [15] and in the present article are similar but not quite identical. It turns out that in the specialization |$t=-q^2$|⁠, the differences can be explained by two different cabling procedures. Our construction for iterated torus knots involved the topological pairs|$({\mathbf{r}},{\mathbf{a}})$| and uses the standard topological cabling |$K^{\mathrm{top}}_{r,a}$| of a knot |$K$| (we refer to |$K^{\mathrm{top}}_{r,a}$| as the topological cable of |$K$|⁠). However, the definition in [15] uses the Newton pairs|$({\mathbf{r}}, {\mathbf{s}})$|⁠, which are related to the topological pairs via equation (A.1). In Section 2.1.5, we described a slightly nonstandard procedure for constructing a cable |$K^{\mathrm{alg}}_{r,s}$| of a framed knot |$K$|⁠, which we call the algebraic cable of |$K$|⁠. Iterating these procedures (starting with the |$0$|-framed unknot), we obtain two framed knots: |$K^{\mathrm{top}}_{{\mathbf{r}},{\mathbf{a}}}$| and |$K^{\mathrm{alg}}_{{\mathbf{r}}, {\mathbf{s}}}$|⁠. It turns out that these two knots are the same, up to an overall framing, and this implies their colored Jones polynomials are equal up to multiplication by a power of |$q$|⁠. We first recall their construction when it is specialized to |${\mathfrak{g}} = {\mathfrak{sl}}_2$| after fixing some notation. Let |${\mathbf{r}} = (r_1,\ldots,r_m)$| and |${\mathbf{s}} = (s_1,\ldots,s_m)$| with |$r_i, s_i \in {\mathbb{Z}}$| relatively prime be Newton pairs, and let |${\mathbf{a}} = (a_1,\ldots,a_m)$| be determined by \begin{equation}\label{eq_newton} a_1 = s_1, \quad a_{i+1} = s_{i+1} + r_i r_{i+1} a_i. \end{equation} (A.1) The algebraic cabling formula of Corollary 2.22 has a simple extension to iterated cables, where in the definition of |$\Gamma_{r,s}^{\mathrm{alg}}$| in (2.5) we identify |$u=y$| as in Remark 5.3. We now recall the definition of the polynomials |$JD_{{\mathbf{r}},{\mathbf{s}}}(n; q,t)$| given in [15, Theorem 2.1] for |${\mathfrak{g}} = {\mathfrak{sl}}_2$|⁠. We will use the notation of Section 4.1. 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Treibich,, Armando

doi: 10.1093/imrn/rnx214pmid: N/A

Abstract Let |$(X,q)$| be a curve of arithmetic genus |$g>0$|⁠, marked at a smooth point and defined over an algebraic closed field. Given |$(n,d)\in \mathbb{N}^\ast\times \mathbb{N,}$| we consider all degree-|$n$| flat generically étale covers |$\pi:\Gamma \to X$|⁠, marked at a subset |$D$| of cardinality |$d$|⁠, satisfying a natural tangency condition inside |$Jac \Gamma$|⁠. We mainly focus in the case |$D\subset \pi^{\,\textrm{-}1}(q)$|⁠, with |$q$| a non-Weierstrass point and |$g\leq d<n$|⁠. All such covers are zero divisors of so-called |$d$|-tangential polynomials and factor through the same ruled surface over |$X$|⁠. Conversely, any generic |$d$|-tangential polynomial gives back such a cover. When |$X$| is a smooth complex curve, we generate all |$d$|-tangential polynomials at once, in terms of the Baker–Akhiezer function of |$(X,q)$|⁠. At last, we consider new phenomena in positive characteristic, namely, infinite towers of Artin–Schreier |$1$|-tangential covers wildly ramified at a unique point. Applications to |$d\times d$|-matrix elliptic (as well as rational and trigonometric) KP solitons are given in the appendix. 1 Introduction Let |$\mathbb{P}^1$| and |$X$| denote, respectively, the projective line and a fixed curve of positive arithmetic genus |$g$|⁠, both defined over an algebraically closed field |$\mathbb{K}$| of arbitrary characteristic |${\boldsymbol{p}}$|⁠. We will fix a local coordinate at a smooth point |$q \in X$|⁠, denoted |$z$|⁠, and consider all generically étale covers |$\pi : \Gamma\rightarrow X$| of fixed degree |$n\geq 1$|⁠. Any such projection can be equipped with a (rational) Abel embedding |$Ab_\Gamma: \Gamma \to Jac\,\Gamma$| into its generalized Jacobian and the morphism |$\pi^* \circ Ab_X: X \to Jac\,\Gamma$|⁠. We identify the tangent space at |$q$|⁠, denoted |$T_{X,q}$|⁠, with |$d(Ab_X\big(T_{X,q})\big) \subset {\rm H}^1(X,O_X)$|⁠. We also identify the tangent space at any smooth point |$r\in \Gamma$| with its image |$d(Ab_\Gamma\big(T_{\Gamma,r})\big) \subset {\rm H}^1(\Gamma,\mathcal{O}_\Gamma)$| in the tangent space at the origin of |$Jac\,\Gamma$|⁠. If |$d(\pi^*)(T_{X,q})= \{0\}$|⁠, which can only happen if |$n\in {\boldsymbol{p}}\mathbb{N} $|⁠, the cover |$\pi$| will be called |$0$|-tangential. Otherwise, for each |$ D \subset \Gamma$| of cardinality |$d>0$| such that, $$ d(\pi^*)(T_{X,q}) \subset \sum_{r\in D} T_{\Gamma,r}\quad\textrm{but }\quad d(\pi^*)(T_{X,q}) \nsubseteq \sum_{r'\in D'} T_{\Gamma,r'}\quad\textrm{for each}\quad D'\nsubseteq D, $$ |$\pi: (\Gamma,D) \to X$| will be called |$D$|-tangential. Whenever |$D\subset \pi^{\,\textrm{-}1}(q),$| we will call it |$d$|-tangential. Such a |$D\subset \pi^{\,\textrm{-}1}(q)$| exists, for example, if |$\pi$| is étale over |$q$|⁠. We will fix |$d<n$| and focus our study on indecomposable |$d$|-tangential covers, that is, those which do not dominate a |$\underline{d}$|-tangential cover with |$\underline{d} <d$|⁠. They are naturally related with irreducible polynomials with coefficients in the function field |$K(X)$|⁠. The latter, so-called |$d$|-tangential, can be easily characterized and give back the original covers as zero divisors of a ruled surface over |$X$|⁠. Both issues are interesting on their own, although they initially came up through the study of KP elliptic solitons and related Calogero-Moser integrable systems (e.g., [1, 4, 5], see also [3, 6, 7] as well as [8] for the KdV elliptic soliton specialization). We proceed as follows: Section 2: We define (minimal/indecomposable) |$D$|-tangential covers and present a |$D$|-tangential criterion characterizing them by the existence of a meromorphic, so-called |$D$|-tangential function. We restrict to |$d$|-tangential ones and give examples of decomposable ones. Section 3: We study the characteristic polynomials of the tangential functions of flat|$d$|-tangential covers. We show that their coefficients are holomorphic outside |$q\in X$| and satisfy affine conditions defining the subvariety |$\theta_{d,n}(X,z)$| of |$d$|-tangential polynomials of degree |$n$|⁠. We find conditions under which |$\theta_{d,n}(X,z)$| is not empty, calculate its dimension and prove its generic element to be irreducible. Section 4: We assume |$X$| smooth and define a ruled surface |$\pi_{{\mathcal S}}: {\mathcal S} \to X$| naturally associated to |$T_{X,q}$|⁠, through which any |$d$|-tangential cover |$\pi: \Gamma \to X$| factors. The indecomposable ones are birational to their image in |${\mathcal S}$|⁠, and their arithmetic genus is bounded by |$(d+g\,\textrm{-}1)n+1\,\textrm{-}\frac{1}{2}d(d+1)$|⁠. At last, we construct a |$(n(d\,\textrm{-}\,g+1)\,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}\,g)(d+g\,\textrm{-}\,1))$|-dimensional family of smooth indecomposable |$d$|-tangential covers of degree |$n$| and genus |$n(d+g\,\textrm{-}1)\,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠. Section 5: We assume |$(X,q)$| is a smooth complex curve of positive genus |$g$| marked at a non-Weierstrass point, introduce its Baker–Akhiezer function and apply it, for any |$d \geq g$|⁠, to the construction of |$\theta_{d}^{\infty}(X,z) \subset K(X)[T] $|⁠, the subring generated by all |$d$|-tangential polynomials. Section 6: In positive characteristic, we fail to provide a global construction for |$\theta_d^{\infty}(X,z) $| but show instead some instances of funny behavior. Notably, the existence of non-trivial |$0$|-tangential polynomials, as well as infinite towers of (wildly ramified Artin–Schreier type) |$1$|-tangential covers. Appendix: We consider a complex rational curve |$X$| with a node or a cusp and obtain polynomial equations for the spectral curves associated to |$d \times d$| matrix KP trigonometric and rational solitons. 2 Tangential Covers By a curve |$C,$| we will always mean an integral projective curve of positive arithmetic genus defined over |$\mathbb{K}$|⁠. We denote |$C^\circ$| the open dense subset of smooth points of |$C$|⁠, |$K(C)$| its function field and |$Jac\, C$| the generalized Jacobian variety of |$C$| (i.e., the moduli space of degree zero line bundles on |$C$|⁠). Its tangent space at the origin is canonically identified with |${\rm H}^1(C, \mathcal{O}_C)$|⁠. Given any |$r\in C^\circ,$| we define the Abel embedding |$Ab_C , C^\circ\hookrightarrow Jac\, C$|⁠, |$r'\mapsto\mathcal{O}_C(r'\textrm{-}\,r)$|⁠. The image of the tangent space to |$C$| at |$r'\in C^\circ$|⁠, denoted by |$T_{C,r'}\subset {\rm H}^1(C,\mathcal{O}_C)$|⁠, is independent of |$r$|⁠. We fix henceforth a projective curve |$X$| of arithmetic genus |$g>0$| marked at a smooth point |$q \in X^\circ$|⁠. We will sometimes assume |$q$| to be a non-Weierstrass point, meaning |${\rm h}^0\big(X,\mathcal{O}_{X}(gq)\big)=1$|⁠. We also recall the following result. Lemma 2.1. (cf., [6, 1.9, p. 615]) Let |$(C,r)$| be a projective curve of positive arithmetic genus marked at a smooth point. Let \begin{align} 0\to\mathcal{O}_C\to\mathcal{O}_C( r)\to\mathcal{O}_{ r}( r)\to 0 \end{align} (1) be the canonical exact sequence and |$\delta, {\rm H}^0(C,\mathcal{O}_{r}(r))\to {\rm H}^1(C,\mathcal{O}_C)$| the coboundary map of the corresponding long exact cohomology sequence. Then |$\delta({\rm H}^0(C,\mathcal{O}_{r}(r)))=T_{C,r}$|⁠. □ Lemma 2.2. Let |$\pi: \Gamma \to X$| be a flat generically étale cover of degree |$n \notin \textbf{p}\mathbb{N}$| and |$\pi^\ast : Jac X \to Jac\Gamma $| the dual homomorphism. Then its derivative at the origin, |$d(\pi^\ast) : {\rm H}^1(X,\mathcal{O}_X) \to {\rm H}^1(\Gamma, \mathcal{O}_\Gamma)$|⁠, is injective. □ Proof. The cover |$\pi$| being flat, we also have the Albanese homomorphism |$Alb(\pi) :Jac \Gamma \to Jac X$|⁠, mapping |$M \in Jac\Gamma$| to the class of |$det(\pi_\ast(M))\otimes det(\pi_\ast(\mathcal{O}_\Gamma))^{\textrm{-}1}$|⁠. Its derivative composed with |$d(\pi^\ast)$| is the multiplication by |$n$|⁠, hence |$d(\pi^\ast)$| is injective. ■ Definition 2.3. A generically étale cover |$\pi : (\Gamma, D)\to X$| marked at |$d$| smooth points |$(d\geq 0 )$| will be called |$D$|-tangential, or simply |$d$|-tangential in case |$D\subset \pi^{\textrm{-}1}(q)$|⁠, if and only if it satisfies one of the following properties: |$d(\pi^\ast)(T_{X,q})=\{0\}$| and |$D=\emptyset$||$($|hence |$d=0); $| |$d(\pi^\ast)(T_{X,q})\neq \{0\}$| and |$D$| is minimal for the following property|$\,:$| \[ d(\pi^\ast)(T_{X,q}) \subset \sum_{r\in D} T_{\Gamma,r}. \] □ Remark 1. For |$d=1$| we get the notion of tangential cover (cf., [6, 1.6]). For |$d>1$|⁠, skipping the minimality condition on |$D$| may give superfluous marked points, meaning that the same projection |$\pi$| could give a |$D'$|-tangential cover for some |$D' \subsetneq D$|⁠. □ Theorem 2.4 (⁠|$D$|-tangency criterion [7, 1.8]). Let |$\pi : (\Gamma,D)\to X$| be a generically étale marked cover and |$z$| a local coordinate at |$q\in X$|⁠. Then |$\pi$| is |$D$|-tangential if and only if |${\rm h}^0(\Gamma,\mathcal{O}_\Gamma(D ))=1$| and there exists a morphism |$\kappa:\Gamma \to \mathbb{P}^1$| such that: |$\kappa$| is holomorphic outside |$D\cup \pi^{\textrm{-}1}(q);$| over a neighborhood of |$D\cup \pi^{\textrm{-}1}(q)$| the divisor of poles of |$\kappa+\pi^\ast(\frac{1}{z})$| is equal to |$D$|⁠. The condition |${\rm h}^0(\Gamma,\mathcal{O}_\Gamma(D))=1$| is equivalent to asking the function |$\kappa$|⁠, called henceforth |$D$|-tangential with respect to |$\pi$|⁠, to be unique up to an additive constant. □ Definition 2.5. Let the covers |$\pi:(\Gamma,D) \to X$| and |$\underline{\pi}:(\underline{\Gamma},\underline{D}) \to X$| be |$D$| and |$ \underline{D}$|-tangential. We will say that |$\pi$| dominates |$\underline{\pi}$|⁠, if and only if there exists a non-trivial morphism |$\varphi:\Gamma \to \underline{\Gamma}$| such that |$\pi=\underline{\pi}\circ \varphi$| and |$\varphi^{\,\textrm{-}1}(\underline{D})=D$|⁠. Conversely, |$\pi$| is called, indecomposable if it does not dominate any |$\underline{D}$|-tangential cover of strictly smaller degree|$;$| minimal, if it does not dominate any other |$\underline{D}$|-tangential cover of same degree. □ Proposition 2.6 (Examples of decomposable |$d$|-tangential covers of degree |$dn$| for any |$1<d \leq n$|⁠). Fix an elliptic curve |$X$|⁠, integers |$1<d \leq n$| and a |$1$|-tangential cover |$\pi: (\Gamma,r) \to X$| of genus and degree |$n$|⁠. Then, for any flat degree-|$d$| cover |$ \varphi : \overline{\Gamma} \to \Gamma$| étale at |$\pi^{\textrm{-}1}(r)$| the morphism |$\pi \circ \varphi :(\overline{\Gamma},\varphi^{\textrm{-}1}(r)) \to X$| is a decomposable |$d$|-tangential cover. □ Proof. Given any tangential function |$\kappa : \Gamma \to \mathbb{P}^1$| one can immediately check that its pull-back |$\varphi^\ast(\kappa)$| satisfies properties (2.4.1 and 2) of the Tangency criterion with respect to |$\pi \circ \varphi $|⁠. In order to complete the proof it only remains to show that |${\rm h}^0\big(\overline{\Gamma},\mathcal{O}_{\overline{\Gamma}}(\varphi^{\textrm{-}1}(r))\big)=1$| and can be done as follows. The moduli space of |$1$|-tangential covers of genus and degree equal to |$n$| is an |$n$|-dimensional affine space (cf., [6, 4.4]). It can also been shown that for any |$j>n,$| there exists |$f_j\in K(\Gamma)$| such that, |$(f_j)_\infty = jr$|⁠. Hence, according to the Weierstrass’s gap Theorem |${\rm h}^0\big(\Gamma,\mathcal{O}_{\Gamma}(nr)\big)=1$|⁠, implying |${\rm h}^0\big(\overline{\Gamma},\mathcal{O}_{\overline{\Gamma}}(\varphi^{\,\textrm{-}1}(r))\big)=1,$| could we find a non-constant morphism |$\overline{\Gamma} \to \mathbb{P}^1$| with |$d'$| poles bounded by |$\varphi^{\,\textrm{-}1}(r)$|⁠, its |$d'$|-th symmetric function with respect to |$\varphi$| would have a pole of order |$d'\leq d \leq n$| at |$r$|⁠. Contradiction! ■ 3 Tangential Polynomials We restrict henceforth to flat |$d$|-tangential covers and study the characteristic polynomials of their |$d$|-tangential functions. All the results can be easily generalized to |$D$|-tangential covers. Definition 3.1. A monic polynomial |$P(T)= T^n + \sum_{j=1}^n\alpha_{j}T^{n\textrm{-}j} \in K(X)[T]$| will be called |$d$|-tangential if and only if it satisfies the following conditions|$:$| for each |$ j=1,\ldots,n$| the coefficient |$ \alpha_{j}$| belongs to |${\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big);$| all coefficients of |$z^dP(T\textrm{-}\,\frac{1}{z})=: z^dT^n + \sum_{j=1}^na_{j}T^{n\textrm{-}j}$| are holomorphic at |$q;$| |$d$| is the least natural satisfying the above property, that is|$,$||$z^dP(T\textrm{-}\,\frac{1}{z})_{|z=0} \neq 0$|⁠. Let |$\theta_{d,n}(X,z) $| denote the subset of |$d$|-tangential polynomials of degree |$n$|⁠. The affine subspace of |$ K(X)[T]$| cut out by the first two conditions is the union |$\Theta_{d,n}(X,z):= \cup_{i=0}^d \theta_{i,n}(X,z)$|⁠. □ Proposition 3.2. Let |$\pi : (\Gamma,D)\to X$| be a |$d$|-tangential cover of degree |$n$| equipped with a |$d$|-tangential function |$\kappa: \Gamma \to \mathbb{P}^1$|⁠, and let |$P_\kappa(T)= T^n + \sum_{j=1}^n\alpha_{j,\kappa}T^{n\textrm{-}j}$| be its characteristic polynomial with respect to the degree |$n$| field extension |$K(\Gamma)/K(X)$|⁠. Then: |$P_\kappa$| belongs to |$\theta_{d,n}(X,z);$| |$P_\kappa(T)$| is irreducible in |$ K(X)[T]$| if |$\pi$| is indecomposable or a power of another one if |$\pi$| is decomposable|$\,;$| let |$\varphi :\overline{\Gamma} \to \Gamma$| denote any morphism of degree |$m\geq 1$| unramified over |$D$| and such that |$\overline{\pi}:=\pi \circ \varphi : \big(\overline{\Gamma},\varphi^{\textrm{-}1}(D)\big)\to X$| is |$md$|-tangential. Then |$\overline{\kappa}:=\kappa \circ \varphi $| is a |$md$|-tangential function for |$\overline{\pi}$| and its characteristic polynomial is equal to |$P_{\overline{\kappa}}(T)=P_\kappa(T)^m$|⁠. □ Proof. (1) Up to a sign |$\alpha_{j,\kappa}$| is the |$j$|-th symmetric function of |$\kappa$| with respect to |$\pi$|⁠. Recall also that |$\kappa$| is holomorphic outside |$\pi^{\textrm{-}1}(q)$| and has at any point |$r \in \pi^{\textrm{-}1}(q)$|⁠, a pole of order bounded by |$ind_\pi(r)$|⁠, the ramification index of |$\pi$| at |$r$|⁠. It follows that |$\alpha_{j,\kappa}\in {\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big)$|⁠. Consider on the other hand |$z^dP_\kappa(T\,\textrm{-}\,\frac{1}{z})= z^dT^n + \sum_{j=1}^na_{j,\kappa}T^{n\textrm{-}j}$|⁠. Then, up to a sign |$a_{j,\kappa}z^{\textrm{-}d}$| is the |$j$|-th symmetric function of |$\kappa +\pi^\ast(\frac{1}{z})$|⁠. The latter has a simple pole at any marked point |$r\in D$| and is holomorphic elsewhere in |$\pi^{\textrm{-}1}(q)$|⁠, hence |$a_{j,\kappa}z^{\textrm{-}d}$| must have a pole at |$q$| of order bounded by |${\rm{min}}\{d,j\}$|⁠. At last, one can check that |$a_{l,\kappa}z^{\textrm{-}d}$| has order |$d$| (at least) for |$l=\sum_{r\in D} ind_\pi(r)$|⁠. (2) and (3)—Straightforward verification. ■ Lemma 3.3. Irreducibility criterion An element |$P= T^n + \sum_j \alpha_jT^{n\textrm{-}j} \in \theta_{d,n}(X,z)$| factors in |$K(X)[T]$|⁠, say |$P=QR$|⁠, if and only if |$Q\in \theta_{d',n'}(X,z)$| and |$R\in \theta_{d\textrm{-}d',n\textrm{-}n'}(X,z)$| for some |$\;0\leq d'\leq d\;$| and |$\;0<n'<n$|⁠. □ Proof. Given |$P=QR \in \theta_{d,n}(X,z),$| we can assume |$Q$| and |$R$| to be monic polynomials of degrees |$n'$| and |$n'':=n\,\textrm{-}\,n'$|⁠, and prove they satisfy the condition |$3.1.1$| as follows. Had |$Q(T,z)$| at least one coefficient with a pole at some point |$q' \neq q$|⁠, for almost any |$t\in \mathbb{K}$| the function |$Q(t,z)$| would also have a pole at |$q'$|⁠. The latter would oblige |$R(t,z)$| to have a zero at |$q'$| for almost any |$t \in \mathbb{K}$| because |$P(t,z)=Q(t,z)R(t,z)$| is holomorphic outside |$ q$|⁠. Hence, the monic polynomial |$R(T,q')$| should have infinitely many roots. Contradiction! An analogous argument applied to the factorization $$ z^nP(z^{\textrm{-}1}T,z)= T^n + \sum_j z^j\alpha_jT^{n\textrm{-}j}=z^{n'}Q(z^{\textrm{-}1}T,z).z^{n''}R(z^{\textrm{-}1}T,z)$$ proves that all coefficients of each factor are holomorphic at |$q$|⁠, hence |$Q$| and |$R$| satisfy |$3.1.1$| as asserted before. At last, there exist |$d'$| and |$d''$| such that |$Q\in \theta_{d',n'}(X,z)$| and |$R\in \theta_{d'',n''}(X,z)$|⁠, hence |$z^{d'+d''}P(T\,\textrm{-}\,\frac{1}{z},z)=z^{d'}Q(T\,\textrm{-}\,\frac{1}{z},z)\cdot z^{d''}R(T\,\textrm{-}\,\frac{1}{z},z)$| must have all its coefficients holomorphic at |$q$| while its restriction to |$z=0$| cannot vanish. Therefore |$d=d'+d''$|⁠. ■ Remark 2. For each |$n>0$| we can immediately check that $$ \Theta_{n,n}(X,z)=T^n+\sum_{j=1}^n {\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big)T^{n\textrm{-j}}. $$ In particular, if |$q\in X$| is non-Weierstrass, meaning |${\rm h}^0\big(X,\mathcal{O}_{X}(gq)\big)=1$|⁠, we have: |$\; {\rm{dim}} \,\Theta_{n,n}(X,z)=n+\frac{1}{2}(n\,\textrm{-}\,g)(n\,\textrm{-}\,g+1)$| for each |$n >g;$| |$\; {\rm{dim}} \,\Theta_{n,n}(X,z)=n$| and any |$P \in \Theta_{n,n}(X,z)$| is completely reducible for each |$n \leq g$|⁠. The next straightforward Lemma implies |$\Theta_{d,n}(X,z)$| is empty in all cases listed below. □ Lemma 3.4. Fix any |$P(T)= T^n + \sum_{j=1}^n\alpha_{j}T^{n\textrm{-}j} \in K(X)[T]$| and let |$z^dP(T\textrm{-}\,\frac{1}{z})=: z^dT^n + \sum_{j=1}^na_{j}T^{n\textrm{-}j}$|⁠. Then for each |$j=1,\ldots,n$| we have $$ a_j(z)= z^d\left(\big(_j^n\big)(\textrm{-}1)^j\frac{1}{z^j}+\sum_{i=1}^j\big(_{j\,\textrm{-}\,i}^{n\,\textrm{-}\,i}\big)(\textrm{-}1)^{j\textrm{-}\,i}\frac{1}{z^{j\textrm{-}\,i}}\alpha_i(z)\right)\!. $$ In case |$q \in X$| is non-Weierstrass, |$g\leq d <n$| and |$\alpha_j \in {\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big)$| for each |$j=1,\ldots,n$|⁠, we also have \begin{align*} a_j(z)& =z^{d\,\textrm{-}j}b_j(z)\;\; \textrm{up to}\; j=d,\textrm{ with}\;b_j\;\textrm{holomorphic such that}\; b_j(0)=(_j^n)(\textrm{-}1)^j\;\; \textrm{if}\;\;j \leq g\\ &\quad{} \textrm{and } b_j(0)=(_j^n)(\textrm{-}1)^j+ \sum_{g<i\leq j}(_{j\,\textrm{-}\,i}^{n\,\textrm{-}\,i})(\textrm{-}1)^{j\textrm{-}\,i}z^i\alpha_i(z)_{|z=0} \quad\textrm{for any}\;\; g<j \leq d. \end{align*} At last asking the remaining coefficients |$\{a_k\,,d<k \leq n \}$| to be holomorphic at |$q$| amounts to solve a triangular system in the polar parts of the Laurent expansions of |$\{\alpha_k \,,d<k \leq n \}$|⁠. The latter system has at each stage a unique solution modulo |${\rm H}^0\big(X,\mathcal{O}_{X}(dq)\big)$|⁠. □ Proposition 3.5. |$\Theta_{0,n}(X,z) = \emptyset,$| in case |$\textbf{p}=0$|⁠, or if |$\textbf{p}>0$| and |$n\notin \textbf{p}\mathbb{N}^\ast.$| |$\Theta_{1,2}(X,z) \neq \emptyset,$| if and only if |${\rm h}^0\big(X,\mathcal{O}_{X}(2q)\big)=2$|⁠. For |$n>2$| and |$g>1,$| we have |$\Theta_{1,n}(X,z)= \emptyset$| each time that, either |$\textbf{p}=0$|⁠, |$\textbf{p}\geq 5$| and |$n(n\textrm{-}1)(n\textrm{-}2) \notin \textbf{p}\mathbb{N}$|⁠, |$\textbf{p}=3$| and |$(n+1)(n+4) \in 9\mathbb{N}$|⁠, or |$\textbf{p}=2$| and |$n+1 \in 4\mathbb{N}$|⁠. If |${\rm h}^0\big(X,\mathcal{O}_{X}(gq)\big)=1$| and |$d<{\rm{min}}\{g,n\}$| then |${\rm{dim}}\big(\Theta_{d,n}(X,z)\big)<n$|⁠. Moreover, |$\Theta_{d,n}(X,z)$| is empty unless |$\textbf{p}>0$|⁠, |$d<g<n$| and |$(_g^n)\in \textbf{p}\mathbb{N}$|⁠. □ Proof. When |$d=0$|⁠, we have |$a_1=\textrm{-}\frac{n}{z}+\alpha_1$|⁠, which should be holomorphic at |$z=0$| forcing |$\alpha_1$| to have a simple pole. Contradiction! When |$d=1,$| we have |$a_2=\frac{1}{z}\,\textrm{-}\,\alpha_1+ z\alpha_2$|⁠, which should be holomorphic at |$z=0$|⁠. Hence |$\alpha_2$| must have a double pole at |$q$|⁠, that is, |$X$| should be hyperelliptic and |$q$| a Weierstrass point. Let |$P(T)= T^n + \sum_{j=1}^n\alpha_{j}T^{n\textrm{-}j} \in \Theta_{1,n}(X,z)$| and |$z^dP(T\textrm{-}\,\frac{1}{z})=: z^dT^n + \sum_{j=1}^na_{j}T^{n\textrm{-}j}$|⁠, with |$n>2$|⁠. Then |$a_2=(_2^n)\frac{1}{z}+(1\,\textrm{-}\,n)\alpha_1+ z\alpha_2$| and |$a_3=\textrm{-}\,(_3^n)\frac{1}{z^2}+(_{\;\;2}^{n\,\textrm{-}\,1})\frac{1}{z}\alpha_1+ (2\,\textrm{-}\,n)\alpha_2+z\alpha_3$|⁠. It follows, assuming the above conditions, that |$\alpha_2$| and |$\alpha_3$| must have a double and triple pole at |$q$|⁠, respectively. Hence |$X$| is an elliptic curve (i.e., |$g=1$|⁠). Suppose |${\rm h}^0\big(X,\mathcal{O}_{X}(gq)\big)=1$| and let |$P(T)= T^n + \sum_{j=1}^n\alpha_{j}T^{n\textrm{-}j} \in K(X)[T]$|⁠, where each |$\alpha_j \in {\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big)$|⁠, and |$z^dP(T\textrm{-}\,\frac{1}{z})= z^dT^n + \sum_{j=1}^na_{j}T^{n\textrm{-}j}$|⁠. Recall that \begin{align*} a_{j} & =\frac{(\textrm{-}1)^{j}}{z^{j\textrm{-}d}}\left((_{j}^{n})+\sum_{i=1}^{j}(^{n\textrm{-}i}_{j\textrm{-}i})(\textrm{-}z)^{i}\alpha_i\right)\quad \textrm{and in particular}\\ a_n(z) & = \frac{(\textrm{-}1)^n}{z^{n\textrm{-}d}}\left(1+\sum_{i=1}^n(\textrm{-}z)^{i}\alpha_i\right)\!. \end{align*} If |$d<n \leq g$|⁠, the coefficient |$a_n(z)$| has a pole of order |$n\,\textrm{-}\,d$| at |$q$|⁠, hence |$\Theta_{d,n}(X,z)=\emptyset$|⁠. Suppose at last |$d<g < n$|⁠. Then |$a_j$| is holomorphic at |$q$| for any |$j\leq d$|⁠, whatever constants |$\{\alpha_1,\cdots,\alpha_j\}$| one may choose. For any other |$j>d$|⁠, once |$\{\alpha_1,\ldots,\alpha_{j\textrm{-}1}\}$| have been chosen there exists at most one element |$\alpha_j \in {\rm H}^0\big(X,\mathcal{O}_{X}(jq)\big)$| modulo an additive constant, such that |$a_j$| is holomorphic at |$q$|⁠. Moreover, their existence implies that |$\big(_j^n\big)\in {\boldsymbol{p}}\mathbb{N}$| for any |$d<j \leq g$|⁠. The latter implies that |${\rm{dim}}\big(\Theta_{d,n}(X,z)\big) \leq n$|⁠. However, for any |$d<j \leq g$|⁠, such |$\alpha_j$| can only exist if |${\boldsymbol{p}}>0$| and |$\big(_j^n\big)\in {\boldsymbol{p}}\mathbb{N}$|⁠. It follows that |$\Theta_{d,n}(X,z)=\emptyset,$| if |${\boldsymbol{p}}=0$| or if there exists |$d<j\leq g$| such that |$\big(_j^n\big)\notin {\boldsymbol{p}}\mathbb{N}$|⁠. At last, since |$ n\,\textrm{-}\,d \geq 2$| and |$a_n$| is holomorphic at |$q$|⁠, we must choose |$\alpha_1=0$|⁠, hence |${\rm{dim}}\big(\Theta_{d,n}(X,z)\big) < n$|⁠. ■ From now on we assume |$g\leq d < n$| and |$q$| non-Weierstrass, but |${\boldsymbol{p}}\geq 0$| unless otherwise stated. Proposition 3.6. Recursive formula |$(\textbf{p}=0)$| Assume |$\textbf{p}=0$| and let |$\Delta, \Delta^{\textrm{-}1}: K(X)[T] \to K(X)[T]$| denote the |$K(X)$|-lineal maps such that \begin{align*} \Delta(T^m)= mT^{m \textrm{-}1} \quad \textrm{and} \quad\Delta^{\textrm{-}1}(T^m)= \frac{1}{m+1}T^ {m+1} \textrm{ for any } m\geq 0. \quad Then, \end{align*} for each |$ P\in \Theta_{d,n}(X,z)$| the polynomial |$\frac{1}{n}\Delta(P) \in K(X)[T]$| belongs to |$\Theta_{d,n\textrm{-}1}(X,z).$| Conversely, for each |$P\in \Theta_{d,n\textrm{-}1}(X,z)$| there exists |$ \phi_P\in {\rm H}^0\big(X,\mathcal{O}_{X}(nq)\big)$|⁠, unique modulo |${\rm H}^0\big(X,\mathcal{O}_{X}(dq)\big)$|⁠, such that, |$n\Delta^{\textrm{-}1}(P)+\phi_P \in \Theta_{d,n}(X,z).$| The map |$\frac{1}{n}\Delta: \Theta_{d,n}(X,z) \to \Theta_{d,n\textrm{-}1}(X,z)$| is therefore surjective with kernel |${\rm H}^0\big(X,\mathcal{O}_{X}(dq)\big).$| □ Proof. Straightforward verification. The local expansion of |$\Delta^{\textrm{-}1}(P)(\frac{1}{z})$| at |$q$| has a pole of order |$\leq n,$| because |$deg\Delta^{\textrm{-}1}(P)=n$|⁠. Thus, |$q$| being non-Weierstrass and |$d\geq g$|⁠, there exists |$\phi_P \in {\rm H}^0\big(X,\mathcal{O}_{X}(nq)\big)$| unique modulo |${\rm H}^0\big(X,\mathcal{O}_{X}(dq)\big)$| such that |$\big(n\Delta^{\textrm{-}1}(P)(\frac {1}{z})+\phi_P(z)\big)_\infty \leq dq$|⁠. It immediately follows that |$n\Delta^{\textrm{-}1}(P)(T)+\phi_P \in \Theta_{d,n\textrm{-}1}(X,z)$|⁠, hence |$\frac{1}{n}\Delta$| is surjective as asserted. ■ Theorem 3.7. The affine space |$\Theta_{d,n}(X,z)$| is not empty and has dimension |$n(d\,\textrm{-}\,g+1)\,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}\,g)(d+g\,\textrm{-}\,1)$|⁠. □ Proof. According to Lemma |$3.4$|⁠, in order for |$P(T)= T^n + \sum_{j=1}^n\alpha_{j}T^{n\textrm{-}j}$| to be in |$ \Theta_{d,n}(X,z)$| the first |$d$| coefficients (i.e., |$\{\alpha_j \in {\rm H}^0(X,\mathcal{O}_X(jq)),\, j \leq d\}$|⁠) can be chosen arbitrarily. As for the last |$n\textrm{-}d$| ones, chosen by solving a triangular system, we can add an arbitrary element in |${\rm H}^0(X,\mathcal{O}_X(dq))$| at each stage, hence the result. When |${\boldsymbol{p}}=0,$| we can give a more enlightening proof as shown hereafter. By differentiating |$n\,\textrm{-}\,d$| times, we get a surjective affine map |$\frac{n!}{(n \textrm{-}\,d)!}\Delta^{n \textrm{-}d}:\Theta_{d,n}(X,z) \to \Theta_{d,d}(X,z)$| with kernel |${\rm H}^0\big(X,\mathcal{O}_{X}(dq)\big)$| at each stage, hence |${\rm{dim}}\,\Theta_{d,n}(X,z) = (n \,\textrm{-}\,d){\rm h}_{d} +\sum_{j=1}^d {\rm h}_{j}$|⁠. ■ Corollary 3.8. The set |$\; \theta_{d,n}(X,z) $| is an open dense subset of |$ \Theta_{d,n}(X,z) $| and its generic element is irreducible. □ Proof. To prove the first assertion it suffices to verify that |$\Theta_{d\textrm{-}1,n}(X,z)$|⁠, the complement of |$\theta_{d,n}(X,z)$| in |$\Theta_{d,n}(X,z)$|⁠, is a proper subspace by comparing their dimensions. If |$d>g,$| we have |${\rm{dim}}\big(\Theta_{d,n}(X,z)\big) \,\textrm{-}\,{\rm{dim}}\big(\Theta_{d\textrm{-}1,n}(X,z)\big)=n+1\,\textrm{-}\,d>0$|⁠, hence the result. For |$d=g$| at last, if |${\boldsymbol{p}}=0$| we know that |$\Theta_{g\textrm{-}1,n}(X,z)= \emptyset$|⁠, hence |$\theta_{g,n}(X,z)=\Theta_{g,n}(X,z)$|⁠. In case |${\boldsymbol{p}}>0$| we have at least |${\rm{dim}}\big(\Theta_{g,n}(X,z)\big)\,\textrm{-}\,{\rm{dim}}\big(\Theta_{g\textrm{-}1,n}(X,z)\big)= n\,\textrm{-}\,{\rm{dim}}\big(\Theta_{g\textrm{-}1,n}(X,z)\big)>0$| according to |$3.5.4)$|⁠. Thus, |$\theta_{g,n}(X,z)$| is always an open dense subset of |$\Theta_{g,n}(X,z)$|⁠. It also follows from |$(3.5.4$|⁠) that |${\rm{dim}}\big(\theta_{d',n'}(X,z)\big)+ {\rm{dim}}\big(\theta_{d\textrm{-}d',n\textrm{-}n'}(X,z)\big) <{\rm{dim}}\big(\theta_{d,n}(X,z)\big)$|⁠, for any |$d'\leq d$| and |$n'<n$|⁠. Thus, according to the irreducibility criterion |$3.3$| the generic element in |$\theta_{d,n}(X,z)$| is irreducible. ■ Remark 3. A slight improvement of the latter result was announced in [9, 1.8], namely, that |$\theta_{g,n}(X,z)=\Theta_{g,n}(X,z)$| and any |$P\in \theta_{g,n}(X,z)$| is irreducible, if |${\boldsymbol{p}}=0$| or if |${\boldsymbol{p}}>0$| and |$(_g^n) \notin {\boldsymbol{p}}\mathbb{N}$|⁠. When |${\boldsymbol{p}}=0$| it is indeed true because |$\Theta_{d',n'}(X,z)=\emptyset$| for any |$d'<g<n'$|⁠, which implies |$\theta_{g,n}(X,z)=\Theta_{g,n}(X,z)$| and that any |$P\in \theta_{g,n}(X,z)$| is irreducible (cf., 3.5.4 and 3.3). However, the existence of non-trivial |$0$|-tangential polynomials when |${\boldsymbol{p}}>0$| implies the existence of reducible polynomials in |$\theta_{g,n}(X,z)$| as soon as |$n \geq {\boldsymbol{p}}^g$||$($|cf., |$6.2)$|⁠. □ 4 Tangential Covers as Divisors of a Ruled Surface We will henceforth assume |$X$| smooth of genus |$g>0$| and identify |$\mathbb{P}^1$| with |$\mathbb{K}\cup\{\infty\}$|⁠. Definition 4.1. Fix a meromorphic function |$z:X \to \mathbb{P}^1$| with a simple zero at |$q$| and |$l$| other ones |$(l\geq 1)$|⁠, say |$\{\alpha_i\}$|⁠, and consider the open affine subsets |$U:=X\setminus \{q\}$| and |$\overline{U}:=X\setminus \{\alpha_i\}$|⁠. Let |$\pi_{{\mathcal S}}: {\mathcal S} \to X$| denote the ruled surface obtained by gluing the fibers of |$\,\mathbb{P}^1\times U\,$| and |$\,\mathbb{P}^1\times \overline{U}\,$| over each |$q' \in U \cap \overline{U} =X\setminus\{q,\alpha_i\}$| by means of a translation as follows|$\,:$| \begin{align*} &\textrm{for any } q' \in U \cap \overline{U,} \,\,\, {\it{we\,\,identify}} \,\,\,(T\,,q') \in \mathbb{P}^1\times U\,\, {\it{with}} \,\,(\overline{T}\,\textrm{-}\, \frac{1}{z(q')}\,,q') \in \mathbb{P}^1\times \overline{U}. \end{align*} The infinity sections |$q' \in U \mapsto (\infty,q')\in \mathbb{P}^1\times U$| and |$q' \in \overline{U} \mapsto (\infty,q')\in \mathbb{P}^1\times \overline{U}$| get glued together and define a particular one denoted by |$C_o \subset S$|⁠. Translating along the fibers of |$\,\mathbb{P}^1\times U$| and |$\,\mathbb{P}^1\times \overline{U}$| by any scalar |$a \in \mathbb{K}$| extends to an automorphism |$\tau_a:{\mathcal S} \to {\mathcal S}$| leaving fixed |$C_o$| and such that |$\pi_{{\mathcal S}} \circ \tau_a =\pi_{{\mathcal S}}$|⁠. Given any |$Q(T)\in K(X)[T]$|⁠, considered as a rational morphism |$\mathbb{P}^1\times U \subset {\mathcal S} \to \mathbb{P}^1$|⁠, the zero divisors |$\{Q(T)=0\} \subset \mathbb{P}^1\times U$| and |$\{Q(\overline{T}\textrm{-}\,\frac{1}{z})=0\} \subset \mathbb{P}^1\times \overline{U} $| get glued over |$U\cap \overline{U}$| and define a divisor in |${\mathcal S}$| denoted by |$Y_Q$|⁠. □ Proposition 4.2. Let |${\mathcal S}_q$| denote the fiber |$\pi_{S}^{\textrm{-}1}(q)$| and |$\kappa_{{\mathcal S}}: {\mathcal S} \to \mathbb{P}^1$| the rational morphism defined by |$T$|⁠. Then|$:$| the canonical divisor of |${\mathcal S}$|⁠, |$K_{{\mathcal S}}$|⁠, is numerically equivalent to |$\textrm{-}2C_o+(2g\textrm{-}2){\mathcal S}_q;$| the section |$C_o$| has genus |$g$| as |$X$| and zero self-intersection|$\,;$| the divisor of zeroes and poles of |$\kappa_{{\mathcal S}}$| is equal to |$Y_T\,\textrm{-} \,(C_o+{\mathcal S}_q);$| the restriction of |$\kappa_{{\mathcal S}}+\pi_{{\mathcal S}}^\ast(\frac{1}{z})$| to |$\mathbb{P}^1\times \overline{U}$| has a simple pole along |$C_o$|⁠. □ Proof. The |$2$|-forms |$dT\wedge dz$| (on |$\mathbb{P}^1\times U$|⁠) and |$d\overline{T}\wedge dz$| (on |$\mathbb{P}^1\times \overline{U}$|⁠) get glued over |$U\cap \overline{U}$| and define, therefore, a meromorphic differential on |${\mathcal S}$| with divisor class as announced. The adjunction formula gives |$g= 1+\frac{1}{2}C_o\cdot(\textrm{-}\,C_o+(2g\,\textrm{-}2){\mathcal S}_q)$|⁠, thus |$ C_o \cdot C_o=0$|⁠. and 4. |$\kappa_{{\mathcal S}}$| is represented over the open subsets |$\mathbb{P}^1\times U$| and |$\mathbb{P}^1\times \overline{U}$| by |$T$| and |$\overline{T}\textrm{-}\frac{1}{z}$|⁠, respectively, hence its divisor of zeros and poles is |$Y_T\,\textrm{-}\, (C_o+{\mathcal S}_q)$|⁠. It also follows that |$\kappa_{{\mathcal S}}+ \pi_{{\mathcal S}}^\ast(\frac{1}{z})$| is represented by |$\overline{T}$| over |$\mathbb{P}^1\times \overline{U}$| and has a simple pole along |$C_o$| as desired. ■ Theorem 4.3. The zero divisor |$Y_P\subset {\mathcal S}$| of any |$P \in \theta_{d,n}(X,z)$| has the following local and global properties|$:$| |$Y_P$| is isomorphic to |$\{P(T)=0\}$| over |$\mathbb{P}^1\times U$| and only intersects |$C_o$| at |$p_{{\mathcal S}}:=C_o\cap {\mathcal S}_q ;$| |$Y_P$| is isomorphic to |$\{z^dP(\overline{T}\textrm{-}\,\frac{1}{z})=0\}$| over |$\mathbb{P}^1\times \overline{U};$| |$Y_P$| has a singularity of multiplicity |$d$| at |$p_{{\mathcal S}}$| with tangent cone transversal to |$C_o;$| |$Y_P$| belongs to |$|nC_o+d{\mathcal S}_q|$|⁠, hence its arithmetic genus is equal to |$n(d+g\,\textrm{-}1)+1\,\textrm{-}\,d$|⁠. □ Proof. (1) and (2). The first two items follow from the construction of |${\mathcal S}$| (cf., |$4.1.4.$|⁠). (3). According to Lemma |$3.4$| all the coefficients of |$z^dP(\overline{T}\,\textrm{-}\,\frac{1}{z})=: z^d\overline{T}^n+ \sum_{j=1}^n a_j\overline{T}^{n\textrm{-}j}$| are holomorphic at |$q$| and |$a_j(z)=z^{d\textrm{-}j}b_j(z)$|⁠, with |$b_j(z)$| holomorphic at |$q$| for any |$j \leq d$|⁠. On the other hand, |$Y_P$| is given over a neighborhood of |$p_{{\mathcal S}}\in {\mathcal S}$| as zero divisor of |$\overline{T}^{\,\textrm{-}n}z^dP(\overline{T}\,\textrm{-}\,\frac{1}{z})= z^d+ \sum_{j=1}^n a_j\overline{T}^{\,\textrm{-}j}$|⁠, hence by |$ z^d+\sum_{j=1}^d b_jz^{d\textrm{-}j}\overline{T}^{\,\textrm{-}j}+\sum_{j>d}^n a_j\overline{T}^{\,\textrm{-}j}$|⁠. It follows that |$Y_P$| has a singularity of order |$d$| at |$p_{{\mathcal S}}$|⁠, with tangent cone transversal to |$C_o$| given by the equation |$z^d+\sum_{j=1}^d b_j(0)z^{d\textrm{-}j}\overline{T}^{\,\textrm{-}j}=0$|⁠. (4). Properties |$4.3.1.$| and |$4.3.3.$| imply |$Y_P \cdot {\mathcal S}_q=n$| and |$Y_P \cdot C_o=d$|⁠, from which follows that |$Y_P\in \vert nC_o+d{\mathcal S}_q\vert$| and has the announced arithmetic genus. ■ Corollary 4.4. Any degree |$n$| indecomposable |$d$|-tangential cover |$\pi:(\Gamma,D) \to X$| factors through a natural morphism |$\iota : \Gamma \to {\mathcal S} $| and dominates a unique minimal |$d$|-tangential cover. Moreover, its arithmetic genus is bounded by |$n(d+g\,\textrm{-}1) \,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠, with equality if and only if |$\pi$| is also a minimal |$d$|-tangential cover and |$\iota(\Gamma)$| has an ordinary singularity at |$p_{\mathcal S}$|⁠. □ Proof. Pick any tangential function |$\kappa: \Gamma \to \mathbb{P}$| corresponding to |$\pi$| and let |$P:=P_{\kappa} \in \theta_{d,n}(X,z)$| denote its characteristic polynomial. The pair |$(\kappa,\pi)$| defines a natural morphism |$\iota:\Gamma \to {\mathcal S}$| (see [7, 2.5, p. 533]) such that |$\iota_*(\Gamma)=Y_P$| and |$\iota^{\textrm{-}1}(p_{{\mathcal S}})=D$|⁠. Any other tangential function corresponding to |$\pi$| is equal to |$\kappa + a$| for an arbitrary |$a\in \mathbb{K}$|⁠, in which case the last morphism should be composed with the automorphism |$\tau_a : {\mathcal S} \to {\mathcal S}$| (see |$4.1.3.$|⁠). If |$\pi$| is indecomposable |$P$| is irreducible (cf., [9]), hence |$Y_P$| is an irreducible curve and |$\iota$| has degree |$1$| on to its image. In particular, |$\iota(\Gamma)=Y_P$| has arithmetic genus equal to |$n(d+g\,\textrm{-}1)\,\textrm{-}d+1$| and a singularity of order |$d$| at |$p_{{\mathcal S}}$|⁠. Consider a sequence of blow-ups |$\varphi: \hat{{\mathcal S}}\to {\mathcal S}$| resolving the singularity |$p_{\mathcal S} \in \iota(\Gamma)$| and let |$\hat{\iota} : \Gamma \to \hat{\Gamma} \subset \hat{{\mathcal S}}$| denote the lift of |$\iota$| to the strict transform of |$\iota(\Gamma)\subset {\mathcal S}$|⁠. The first blow-up of |$p_{{\mathcal S}}\in \iota(\Gamma)\subset {\mathcal S}$| drops its arithmetic genus by |$\frac{1}{2}d(d\,\textrm{-}1)$|⁠. Therefore, the arithmetic genus of |$\hat{\Gamma}$| is bounded by |$n(d+g\,\textrm{-}1) \,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠, with equality if and only if |$\iota(\Gamma)$| has an ordinary singularity at |$p_{\mathcal S}$|⁠. We also remark that the birational morphism |$\hat{\iota}$| composed with the |$d$|-marked projection |$\hat{\pi}:=\pi_{\mathcal S} \circ \varphi:\big(\hat{\Gamma},\varphi^{\,\textrm{-}1}(p_{\mathcal S})\big) \to X$| and the function |$\hat{\kappa}:= \kappa_{{\mathcal S}} \circ \varphi : \Gamma \to \mathbb{P}^1$| gives back the pair |$(\pi,\kappa)$|⁠. It follows that |$(\hat{\pi},\hat{\kappa})$| satisfies the Tangency criterion |$2.4$|⁠; that is, |$\hat{\pi}$| is a |$d$|-tangential cover. Any other |$d$|-tangential cover dominated by |$\pi$| can also be equipped with a tangential function dominated by the above function |$\kappa$| and has same natural image |$\iota(\Gamma)$| in |${\mathcal S}$|⁠. In other words |$\hat{\pi}$| is the unique minimal |$d$|-tangential cover dominated by |$\pi$|⁠. We deduce at last that the arithmetic genus of |$\Gamma$| is bounded by |$n(d+g\,\textrm{-}1) \,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠, with equality if and only if |$\pi$| is minimal (i.e., |$\Gamma=\hat{\Gamma}$|⁠) and |$\iota(\Gamma)$| has an ordinary singularity at |$p_{\mathcal S}$|⁠. ■ Definition 4.5. Let |$P(T) \in \Theta_{d,n}(X,z)$| and |$z^dP(T\textrm{-}\,\frac{1}{z})= z^dT^n + \sum_{j=1}^n a_{j}T^{n\textrm{-}j}$| as above. Recall |$(3.4)$| that the coefficients |$\{a_j,\,j \leq n\}$| are holomorphic at |$q $| and |$a_j(z)=z^{d\textrm{-}j}b_j(z)$| for any |$j=1,\ldots,d$| with |$b_j$| holomorphic at |$q$|⁠. We assign to |$P(T) \in \Theta_{d,n}(X,z)$|⁠: $$ V_{d,n}(P):=z^dP \left(T\textrm{-}\,\frac{1}{z}\right)_{|z=0} =\sum_{j=d}^n a_{j}(0)T^{n\textrm{-}j}\quad \textrm{as well as} \quad M_{d,n}(P):=w^d+\sum_{j=1}^d b_j(0)w^{d\textrm{-}j} $$ and let $$V_{d,n}: \Theta_{d,n}(X,z)\to \mathbb{K}_{n\textrm{-}d}\left[\,T\,\right]\quad \textrm{and}\quad M_{d,n}: \Theta_{d,n}(X,z) \to \mathbb{K}_{d}\left[\,w\,\right]$$ denote the corresponding affine linear mappings. □ Lemma 4.6. Assume |$q\in X$| non-Weierstrass and |$g < d < n$|⁠. Then |$M_{d,n}(P)$| and |$V_{d,n}(P)$| have |$d$| and |$n\,\textrm{-}\,d$| simple non-zero roots, respectively, for a generic |$P \in \Theta_{d,n}(X,z)$|⁠. A similar result holds for |$M_{g,n}(P)$| and |$V_{g,n}(P)$|⁠, provided |$(n\,\textrm{-}\,g)\big(_g^n\big) \notin \textbf{p}\mathbb{N}$|⁠. □ Proof. In each case, it is enough to construct a polynomial satisfying all properties. For |$\Theta_{d,n}(X,z)$| with |$g<d<n$| it follows directly from Lemma |$3.4$| but a nicer proof exists when |${\boldsymbol{p}}=0$|⁠. In the latter case, there exists |$f_d\in {\rm H}^0\big(X,O_X(dq)\big)$| such that |$f_d(z)=\frac{1}{z^d}\big(1+O(z)\big)$|⁠, and the affine map $$\frac{1}{n}\Delta:\Theta_{d,n}(X,z) \to \Theta_{d,n\textrm{-}1}(X,z), \quad \textrm{as well as }\quad \frac{(n\,\textrm{-}\,d)!}{n!}\Delta^{(n\textrm{-}d)}:\Theta_{d,n}(X,z) \to \Theta_{d,d}(X,z)$$ is surjective |$($|see |$3.6)$|⁠. Hence, for each |$P \in \Theta_{d,n}(X,z)$| with |$M_{d,n}(P)=w^d+\sum_{j=1}^d b_j(0)w^{d\textrm{-}j}$|⁠, we obtain |$M_{d,n\textrm{-}1}\big(\frac{1}{n} \Delta(P)\big)=w^d+\sum_{j=1}^d b_j(0)\frac{(n\,\textrm{-}\,j)}{n}w^{d\textrm{-}j}$|⁠, and after |$n\,\textrm{-}\,d$| differentiations|$:$| $$ M_{d,d}\left(\frac{(n\,\textrm{-}\,d)!}{n!} \Delta^{(n\textrm{-}d)}(P)\right) = w^d+\frac{1}{(_d^n)}\sum_{j=1}^d b_j(0)(^{n\,\textrm{-}\,j}_{d\,\textrm{-}\,j})w^{d\textrm{-}j}. $$ For any |$a\in \mathbb{K}$| let |$P_a\in \Theta_{d,n}(X,z)$| denote a pre-image of |$R_a:=T^d+af_d \in \Theta_{d,d}(X,z)$|⁠. The above equality relates |$M_{d,d}(R_a)= (w\,\textrm{-}1)^d+a$| with |$M_{d,n\,\textrm{-}1}\big(\frac{1}{n} \Delta(P)\big)$| and implies that |$M_{d,n}(P_a)=M_{d,n}(P_0)+a(_d^n).$| In particular, |$M_{d,n}(P_a)$| has |$d$| non-zero simple roots for almost any |$a\in \mathbb{K}$| and |$deg\big(V_{d,n}(P_a)\big)=n\,\textrm{-}\,d$|⁠. We can also check that |$P_a+bT^d \in \Theta_{d,n}(X,z)$| for each |$b\in \mathbb{K}$| and |$M_{d,n}(P_a +bT^d)=M_{d,n}(P_a)$|⁠, as well as the equality |$V_{d,n}(P_a +bT^d)=V_{d,n}(P_a)+ (\textrm{-}1)^db$|⁠. It follows that for generic |$a,b\in \mathbb{K}$| the polynomial |$P_a +bT^d$| has the required properties. As for the remaining case (i.e., |$d=g$|⁠), we present hereafter a proof valid in arbitrary characteristic. Given any |$P\in \Theta_{g,n}(X,z),$| we must prove that |$M_{g,n}(P)(w)= w^g+\sum _{j=1}^g (\,\textrm{-}\,1)^j(_j^n)w^{n\,\textrm{-}\,j}$|⁠, or equivalently |$R(x):= 1+\sum _{j=1}^g (_j^n)(\,\textrm{-}\,x)^j$|⁠, has |$g$| simple roots. It suffices to check that |$R$| and its derivative |$R'$| have no common root. The latter satisfy $$(1\,\textrm{-}\,x)^n=R(x)+ (^{\;n\,}_{g+1})(\textrm{-}x)^{g+1}+ O(x^{g+2})\quad \textrm{and} \quad \textrm{-}n(1\,\textrm{-}\,x)^{n\,\textrm{-}\,1}=R'(x)\,\textrm{-}\,(g+1)(^{\;n\,}_{g+1})(\textrm{-}x)^{g}+ O(x^{g+1}),$$ implying |$\textrm{-}nR(x)=(1\,\textrm{-}\,x)R'(x)\,\textrm{-}\, (g+1)(^{\;n\,}_{g+1})(\textrm{-}x)^{g}$|⁠. Thus, assuming |$(g+1)(^{\;n\,}_{g+1})=(n \textrm{-}g)(^n_g) \neq 0$| implies |$n\neq0$|⁠, hence, |$R(x)$| and |$R'(x)$| have no common root. ■ Proposition 4.7. Assume |$q\in X$| non-Weierstrass and |$g \leq d < n$|⁠, with |$(n\textrm{-}g)\big(_g^n\big) \notin \textbf{p}\mathbb{N}$| if |$d=g$|⁠. Then for a generic |$P \in \theta_{d,n}(X,z):$| |$Y_P$| has an ordinary singularity at |$p_{{\mathcal S}}$| of multiplicity |$d$| and transversal to |$ C_o+{\mathcal S}_q;$| |$Y_P$| is irreducible, and also smooth outside |$p_{{\mathcal S}}$| if |$\textbf{p}=0$|⁠. □ Proof. The tangent cone of |$Y_P$| at |$p_{{\mathcal S}}$| is the zero locus of the degree-|$d$| form |$\overline{T}^{\,\textrm{-}\,d}M_{d,n}(P)(z\overline{T})$|⁠. For generic |$P$| is the union of |$d$| lines transversal to |$C_o+{\mathcal S}_q$| (see |$3.4$|⁠, the proof of |$4.3.3.$| and |$4.6$|⁠). According to Corollary |$3.8$|⁠, |$P$|⁠, hence |$Y_P$|⁠, is irreducible. Asking |$Y_P$| to be smooth at every point in |${\mathcal S}_q \setminus\{p_{{\mathcal S}}\}$| or outside |${\mathcal S}_q$| are open conditions on |$\theta_{d,n}(X,z)$|⁠. The first one is true as soon as |$V_{d,n}(P)$| has |$n\,\textrm{-}\,d$| simple roots. As for the second one, we can pick |$Y_{P}$| smooth over |$q$| and argue as follows. The curves |$Y_{P}$| and |$Y_{P+1}$| are disjoint outside |${{\mathcal S}}_q$| and generate the affine pencil |$\{Y_{P+a},\,a\in \mathbb{K}\}$|⁠. When |${\boldsymbol{p}}=0,$| Bertini’s Theorem applies, according to which |$Y_{P+a}$| is also smooth over |$X\setminus\{q\}$| for almost any |$a\in \mathbb{K}$|⁠. ■ Theorem 4.8. Assume |$q\in X$| non-Weierstrass and |$g \leq d < n$|⁠, with |$(n\textrm{-}g)\big(_g^n\big) \notin \textbf{p}\mathbb{N}$| if |$d=g$|⁠. Pick a generic |$P \in \theta_{d,n}(X,z)$| and let |$e:\hat{Y}_P\subset \hat{{\mathcal S}} \to Y_P \subset {\mathcal S}$| denote the strict transform of |$Y_P$| by the blow-up of |$p_{{\mathcal S}} \in {\mathcal S}$|⁠. Then |$\pi:= \pi_{{\mathcal S}} \circ e :\hat{Y}_P \to X$| is étale at the |$d$| pre-images |$\{ \hat{r}_i\}:=e^{\textrm{-}1}(p_S) \subset \pi^{\textrm{-}1}(q)$| and|$\,:$| |$\hat{Y}_P$| has arithmetic genus |$n(d+g\,\textrm{-}1) \,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$| and is smooth if |$\textbf{p}=0;$| |$\kappa:=\kappa_{{\mathcal S}} \circ e$| is holomorphic outside |$\pi^{\textrm{-}1}(q)$| and |$\kappa+\pi^\ast(\frac{1}{z})$| has pole divisor |$\sum_{i=1}^d \hat{r}_i;$| |$\pi :(\hat{Y}_P,\sum_{i=1}^d \hat{r}_i) \to X$| is a minimal indecomposable |$d$|-tangential cover. □ Proof. The first two items follow immediately from the preceding properties. As for the third item, it is enough to prove that |$\pi$| is |$d$|-tangential. Should it be true, |$\kappa$| would be a |$d$|-tangential function for |$\pi$| and |$P(T)$| its characteristic polynomial. Since |$P(T)$| is irreducible and the curve |$\hat{Y}_P$| has genus |$n(d+g\,\textrm{-}1) \,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠, we would deduce from Proposition |$3.2.3$| and Corollary |$4.4$| that |$\pi$| is also indecomposable and minimal. Let us recall that |${\rm{dim}}\, \Theta_{d,n}(X,z) \,\textrm{-}\, {\rm{dim}}\, \Theta_{d\,\textrm{-}1,n}(X,z)=n\,\textrm{-}\,d+1 \geq 2$| (see |$3.7$|⁠) before completing the proof. According to the Tangential criterion |$2.4$| it only remains to check the equality |${\rm h}^0(\hat{Y}_P,\mathcal{O}_{\hat{Y}_P}(\sum_i \hat{r}_i))=1$| for generic |$P$|⁠. Assume on the contrary that, |${\rm h}^0(\hat{Y}_P,\mathcal{O}_{\hat{Y}_P}(\sum_i \hat{r}_i))>1$| for a generic |$P \in \theta_{d,n}(X,z)$|⁠. Then for each non constant |$h\in {\rm H}^0(\hat{Y}_P,\mathcal{O}_{\hat{Y}_P}(\sum_i \hat{r}_i))$| there exists at least one |$\lambda \in \mathbb{K}$| such that |$\kappa+\lambda h+\pi^\ast(\frac{1}{z})$| has less than |$d$| poles. It would follow that the characteristic polynomials of |$\{\kappa+ \lambda h, \,\lambda \in \mathbb{K}\}$| define a one dimensional family in |$\Theta_{d,n}(X,z)$| with at least one element in |$\Theta_{d\,\textrm{-}1,n}(X,z)=\bigcup_{k=1}^{d\,\textrm{-}1} \theta_{k,n}(X,z)$|⁠. This cannot happen if |$d=g$| because |$(_g^n) \notin {\boldsymbol{p}}\mathbb{N}$| implies |$\Theta_{g\,\textrm{-}1,n}(X,z)=\emptyset$| (see |$3.5.4))$|⁠. In case |$d>g$| all such polynomials make up a subvariety of dimension bounded by |${\rm{dim}}\Theta_{d\,\textrm{-}1,n}(X,z)+1$|⁠, strictly smaller than |${\rm{dim}}\, \Theta_{d,n}(X,z),$| and we still get a contradiction. ■ Corollary 4.9. Assume |$q\in X$| non-Weierstrass and |$g \leq d < n$|⁠, with |$(n\textrm{-}g)\big(_g^n\big) \notin \textbf{p}\mathbb{N}$| if |$d=g$|⁠. Then there exists a family of indecomposable and minimal |$d$|-tangential degree-|$n$| covers of |$(X,q)$|⁠, of dimension |$n(d\,\textrm{-}\,g+1)\,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}\,g)(d+g\,\textrm{-}\,1),$| and genus |$n(d+g\,\textrm{-}1)\,\textrm{-}\,\frac{1}{2}(d\,\textrm{-}1)(d+2)$|⁠. The latter are smooth if |$\textbf{p}=0$|⁠. □ 5 Tangential Polynomials via the Baker–Akhiezer Function We restrict in this section to a smooth complex curve of genus |$g>0$|⁠, equipped with a local coordinate |$z$| at a non-Weierstrass point |$q$|⁠. We will freely follow the notations and results developed in Dubrovin’s remarkable survey [3], recalling the so-called Baker–Akhiezer function with essential singularity of type |$e^{\frac{x}{z}}$| at |$q$| and pole divisor |$gq$|⁠, from which we will obtain all |$d$|-tangential polynomials at once. After suitable choices of basis for the integral homology and the vector space of holomorphic differentials, the Abel map |$A_X:(X,q) \to (Jac\,X,0)$| is reinterpreted as an embedding in a complex torus |$\mathbb{C}^g/\Lambda$|⁠. Let |$\vartheta_X:\mathbb{C}^g \to \mathbb{C}$| denote the corresponding holomorphic |$\Lambda$|-multiplicative theta function, |$\Theta_X \subset \mathbb{C}^g/\Lambda$| denote its zero divisor and |$K\in \mathbb{C}^g$| be the vector of Riemann constants. Jacobi’s inversion Theorem in this set up asserts that any non-special effective degree-|$g$| divisor |$D$| on |$X$| is the zero divisor of the holomorphic many-valued function |$\vartheta_X\big(A_X(z)\,\textrm{-}\,A_X(D)\,\textrm{-}\,K\big)$|⁠. For |$D=gq$| it says that |$A_X(D)=0$| and |$\vartheta_X\big(A_X(z)\,\textrm{-}\,K\big)=cz^g(1+O(z))$| for some |$c\in \mathbb{C}^\ast$|⁠, meaning geometrically that the translated hypersurface |$K + \Theta_X$| and the curve |$A_X(X)$| intersect (uniquely) at |$A_X(q)=0$| with multiplicity |$g$|⁠. We also introduce |$\omega^{(1)}_q$|⁠, the unique meromorphic differential of the second kind with zero |$a$|-periods and local expansion |$\omega^{(1)}_q=\frac{(1+O(z))}{z^2}dz$| at |$q$|⁠. Its vector of |$b\,$|-periods, say |$U\in \mathbb{C}^g$|⁠, is equal to |$\partial_z A_X(z)_{\vert z=0}$| and is naturally identified with the tangent vector |$\delta([\frac{1}{z}])\in T_{X,q}$| (cf., Lemma |$2.1$|⁠). Integrating along the same path as for the Abel map |$A_X,$| we obtain the many-valued meromorphic function |$\zeta(z):= \textrm{-}\,\int_q^z \omega^{(1)}_q$|⁠, well defined over the universal covering of |$X$|⁠. Taking into account its properties, as well as those of |$F(z,x):=\vartheta_X\big(A_X(z)\,\textrm{-}\,K+xU\big)$|⁠, gives the following explicit formula for the Baker–Akhiezer function |$\psi_{gq}$| corresponding to the non-special divisor |$gq$|⁠. Theorem 5.1 (cf., [3, p. 45]). For each |$x\in \mathbb{C}$| the function |$\psi_{gq}(z,x):=e^{x\zeta(z)}\frac{F(z,x)}{F(z,0)}$| is well defined on |$X$| and holomorphic outside |$q$|⁠, where it has the local expansion |$\psi_{gq}(z,x)=e^{\frac{x}{z}}\frac{1}{z^g}O(1)$|⁠. □ Once |$x$| is formally replaced by |$\Delta:= \frac{\partial}{\partial T}$| in the Maclaurin expansion of |$\psi_{gq}(z,x),$| we can make it operate on |$\mathbb{C}[T]$|⁠. The outcome is an isomorphism for each |$n$| between |$M_n$|⁠, the degree-|$n$| monic polynomials, and |$\Theta_{g,n}(X,z)$| (cf., [6] for |$g=1$|⁠). To prove it, along with similar characterizations for any |$n> d\geq g \geq 1$|⁠, we will need the following properties of |$F(z,x):=\vartheta_X\big(A_X(z)\,\textrm{-}\,K+xU\big)$| and the |$k$|-th partial derivatives of |$\psi_{gq}(z,x)$| as functions of the variable |$x$|⁠. Lemma 5.2. The Maclaurin series in |$x$| of |$\psi(z,x):= \psi_{gq}(z,x)$| and |$F(z,x)$| have the following properties|$:$| |$\psi(z,x):= 1+\sum_{m=1}^\infty \alpha_mx^m$| with |$\alpha_m \in {\rm H}^0\big(X,\mathcal{O}_{X}(mq\big)$| for each |$m$|⁠, hence constant if |$m\leq g;$| the zero locus of |$F(z,x)$| has a singularity of multiplicity |$g$| at the origin with tangent cone|$\,:$| \begin{align*} \left\{\sum_{j=0}^g\frac{(\textrm{-}1)^j}{j!}z^{g\textrm{-}j}x^j=0\right\}\!; \end{align*} for each |$m > g$| the coefficient |$\alpha_m$| has a pole of order |$m$| at |$q$| and \begin{align*} z^m\alpha_{m \vert z=0}=\frac{1}{m!}\sum_{k=0}^g(\textrm{-}1)^k (_{\,k}^m)=(\textrm{-}1)^g\frac{1}{m!}(_{\;\;g}^{m\textrm{-}1}); \end{align*} for each |$m\geq 1$| the |$m$|-th partial derivative of |$\psi(z,x)$| with respect to |$x$| is equal to $$ \psi^{(m)}:=\partial_x^m\big(\psi(x,z)\big)=\frac{e^{x\zeta(z)}}{m!F(z,0)} \left(\sum_{j=0}^m(_{\,j}^m)\zeta^{m\textrm{-}j}\partial_x^jF\right) = m!\alpha_m+O(x) $$ and satisfies: (a) |$z^{m+g}e^{\textrm{-}\frac{x}{z}}\psi^{(m)}=\sum_{j=0}^\infty \beta_jx^j$| has all its coefficients holomorphic at |$q;$| (b) its restriction to |$z=0$| satisfies |$z^{m+g}e^{\textrm{-}\frac{x}{z}}\psi^{(m)} _{\vert z=0}=\frac{(\textrm{-}x)^g}{m!g!}\big(1+O(x)\big)$|⁠. □ Proof. (1) and (2) Expanding |$\psi(z,x):=e^{x\zeta}\frac{F(z,x)}{F(z,0)}=(1+\sum_j \frac{1}{j!}\zeta^jx^j)\big(1+\sum_i\frac{\partial_x^iF}{F}(z,0)x^i\big)$| with respect to |$x$| we obtain the following formula: $$m!\alpha_m=\zeta^m+ \sum_{k=1}^m (^m_{\,k})\zeta^{m\,\textrm{-}\,k}\frac{\partial_x^kF}{F}(z,0).$$ The coefficients |$\{\alpha_m, m>0\}$| must be well defined on |$X$| and holomorphic outside |$q$| just as |$\psi(z,x)$| itself for each |$x$|⁠. For example, |$\alpha_1=\zeta(z)+\frac{\partial_xF}{F}(z,0)$|⁠, implying it has a pole divisor bounded by |$gq$|⁠. Therefore |$\alpha_1$| is constant and |$\partial_xF(z,0)=\textrm{-}\,cz^{g\,\textrm{-}\,1}\big(1+O(z)\big)$|⁠, because |${\rm h}^0\big(X, \mathcal{O}_X(gq)\big)=1$| and |$F(z,0)=cz^{g}\big(1+O(z)\big)$|⁠. Analogously, as long as |$m\leq g$|⁠, it also implies that |$\alpha_m$| is constant and |$\partial_x^mF(z,0)=(\textrm{-}\,1)^m\,cz^{g\,\textrm{-}\,m}\big(1+O(z)\big)$|⁠, hence the zero locus of |$F(z,x)$| has the announced singularity. (3) For each |$m >g$| instead, a direct calculation gives $$z^m m!\alpha_m(z)=\sum_{k=0}^g(\textrm{-}1)^k (_{\,k}^m)+O(z)=(\textrm{-}1)^g(_{\;\;g}^{m\,\textrm{-}1})+O(z) \textrm{ which implies} \quad (\alpha_m)_\infty =mq.$$ (4) Recall that |$z\zeta(z)$|⁠, |$\zeta(z)\,\textrm{-}\,\frac{1}{z}$| and |$\frac{z^g}{F(z,0)}$| are holomorphic in a neighborhood of |$q$|⁠, with values at |$q$| equal to |$1$|⁠, |$a$| and |$\frac{1}{c}$| for some |$c\in \mathbb{C}^\ast$| and |$a\in \mathbb{C}$|⁠. Therefore, the Maclaurin expansion in |$x$| of $$z^{m+g}e^{\textrm{-}\frac{x}{z}}\psi^{(m)}=\frac{e^{x(\zeta\,\textrm{-}\,\frac{1}{z})}}{m!}\frac{z^g}{F(z,0)}\left(\sum_{j=0}^m(_{\,j}^m)(z\zeta)^{m\textrm{-}j}z^j \partial_x^jF\right) $$ has all its coefficients holomorphic at |$q$|⁠. It also follows that its restriction to |$z=0$| is equal to |$\frac{e^{ax}}{m!c}F(0,x)=\frac{(\textrm{-}x)^g}{m!g!}\big(1+O(x)\big)$| as already shown in this proof. ■ Lemma 5.3. For each |$n \geq g \geq 1$| the map |$\psi:= \psi(z,\Delta): \mathbb{C}[T] \to K(X)[T]$| identifies |$M_n$| with |$ \theta_{g,n}(X,z)$|⁠. Given any |$n\geq d > g \geq 1$| and |$P\in \Theta_{d,n}(X,z)$| there exists a unique |$R\in T^g\mathbb{C}_{n\textrm{-}d}[T]$| such that \[ P\,\textrm{-}\,\psi^{(d\textrm{-}\,g)}(R) \in \Theta_{d\,\textrm{-}1,n}(X,z). \] □ Proof. Recall that |$\Theta_{g\,\textrm{-}1,n}(X,z)=\emptyset$| (see |$3.5.4)$|⁠), which implies |$\theta_{g,n}(X,z)=\Theta_{g,n}(X,z)$|⁠. Coupling the formula |$e^{\textrm{-}\frac{\Delta}{z}}\big(Q(T)\big)=Q(T\,\textrm{-}\,\frac{1}{z})$|⁠, valid for each |$Q(T) \in K(X)[T]$|⁠, with the latter properties of |$\psi:=\psi(z,\Delta)$| and its |$k$|-th partial derivatives |$\psi^{(k)}:=\partial_x^k\psi(z,\Delta),$| we then prove that: \begin{align*} & \textrm{for each } M \in M_n\; \textrm{the polynomials}\; P:=\psi(M)\quad \textrm{and} \quad z^gP\left(T\,\textrm{-}\,\frac{1}{z}\right)\\ &\quad{} = z^ge^{\Delta(\zeta\,\textrm{-}\,\frac{1}{z})}\frac{F(z,\Delta)}{F(z,0)}(M) \end{align*} satisfy properties (3.1.1 and 2), that is, |$P \in \Theta_{g,n}(X,z)=\theta_{g,n}(X,z)$|⁠. On the other hand, the map |$ \psi: \mathbb{C}[T] \to K(X)[T]$| preserves degrees and restricts to an injection between subspaces of same dimension: |${\rm{dim}} M_n=n= {\rm{dim}} \Theta_{g,n}(X,z)$|⁠, hence it is an isomorphism as desired. We can also prove by means of the latter results that for some |$b\neq0$| and any |$R\in \mathbb{C}_{n\textrm{-}d+g}[T]$|⁠: (a) |$Q:=\psi^{(d\textrm{-}g)}(R) $| has same leading term as |$(d\,\textrm{-}\,g)!\alpha_{d\,\textrm{-}g}R\,$| and degree |$\leq n\,\textrm{-}(d\,\textrm{-}\,g)<n;$| (b) all coefficients of |$z^dQ(T\,\textrm{-}\,\frac{1}{z})=z^de^{\textrm{-}\frac{\Delta}{z}}\psi^{(d\textrm{-}g)}(R)=(\sum_j\beta_j\Delta^j)(R)$| are holomorphic at |$q$|⁠; (c) |$z^dQ(T\,\textrm{-}\,\frac{1}{z})_{\vert z=0} =\big(b+O(\Delta)\big)\Delta^g(R)\in \mathbb{C}_{n\textrm{-}d}[T]\;($|see |$5.2.4.b)\,)$|⁠; (d) the linear map |$R\in \mathbb{C}_{n\textrm{-}d+g}[T] \mapsto z^dQ(T\,\textrm{-}\,\frac{1}{z})_{\vert z=0} \in \mathbb{C}_{n\textrm{-}d}[T]$| has kernel |$\mathbb{C}_{g\,\textrm{-}1}[T]$|⁠. Therefore, there exists a unique |$R\in T^g\mathbb{C}_{n\textrm{-}d}[T]$| such that |$\big(P\,\textrm{-}\,\psi^{(d\textrm{-}g)}(R)\big)_{\vert z=0}=0$|⁠, from which follows that |$P\,\textrm{-}\,\psi^{(d\textrm{-}g)}(R)\in \Theta_{d\,\textrm{-}\,1,n}(X,z)$|⁠. ■ By repeatedly applying the last lemma, we obtain the following theorem. Theorem 5.4. For each |$n\geq d >1` g \geq 1$| we have |$\Theta_{g,n}(X,z)= \psi(M_n)$| and|$\,:$| \[ \Theta_{d,n}(X,z)= \psi(M_n)\oplus \bigoplus_{k=1}^{d\,\textrm{-}\,g}\psi^{(k)}(T^g\mathbb{C}_{n\textrm{-}g\textrm{-}k}[T]). \] □ 6 Towers of Artin–Schreier |$1$|-Tangential Covers (⁠|${\boldsymbol{p}}>0$|⁠) We restrict at last to the positive characteristic case and obtain new phenomena. The key role is played by |$0$|-tangential polynomials of the form |$A(T)+f$| with |$A(T) \in \mathbb{K}[T]$| an additive polynomial, that is, such that |$A(T+a)=A(T)+A(a),$| for any |$a\in \mathbb{K}$|⁠. We do not assume the marked point |$q \in X$| to be non-Weierstrass. Definition 6.1. To any local coordinate |$z$| at |$q\in X$| we associate the tangent vector |$t:=\delta([\frac{1}{z}])\in T_{X,q}$||$($|cf., |$2.1)$| and let |$\overline{Orb}_\textit{F}(q)\subset {\rm H}^1(X,O_X)$| denote the subspace generated by its orbit |$\{\delta([\frac{1}{z^{p^j}}]),\;0\leq j\}$| under the Frobenius action. Its dimension, denoted |$m_q:= dim\overline{Orb}_\textit{F}(q)$|⁠, depends on the point |$q$| but is bounded by |$g$|⁠. □ Lemma 6.2. There exists a unique monic additive polynomial |$A_t(T) \in \mathbb{K}[T]$| of degree |$p^{m_q}$| and function |$f_t\in {\rm H}^0\big(X,O_X(p^{m_q}q)\big)$| satisfying |$f_t(z)=A_t(\frac{1}{z})+O(z)$|⁠. Their sum |$P_t(T):=A_t(T)+f_t$| is an irreducible |$0$|-tangential polynomial and the ring homomorphism $$ B(T)\in \mathbb{K}[T] \mapsto B(P_t(T)) \in K(X)[T] $$ sets an isomorphism between |$\mathbb{K}[T]$| and |$\theta_{0}^\infty(X,z)$|⁠, the subring generated by the set of all |$0$|-tangential polynomials. Moreover, |$P(T) \in\theta_{0}^\infty(X,z)$| decomposes as |$A(T)+f$| with |$A(T)\in \mathbb{K}[T]$| if and only if there exists an additive |$B \in \mathbb{K}[T]$| satisfying |$B(P_t(T))=P(T)$|⁠. In particular, up to a constant |$A=B(A_t)$| is also additive and |$f=B(f_t)$|⁠. □ Proof. Let |$m$| denote the maximal positive integer such that |$\{\delta([\frac{1}{z^{p^j}}]),\;0\leq j<m\}$| is linearly independent. The latter family generates a subspace contained in |$\overline{Orb}_F(q)$| and invariant by the Frobenius action, hence they are equal and |$m=m_q$|⁠. It also implies that |$\{\delta([\frac{1}{z^{p^j}}]),\;0\leq j<m\}$| is a basis of |$\overline{Orb}_F(q)$| and there exists a unique linear relation |$\delta\big([A_t(\frac{1}{z})]\big)=0 \in {\rm H}^1(X,O_X)$|⁠, hence a function |$f_t$| as asserted. The polynomial |$P_t(T):=A_t(T)+f_t$| is therefore |$0$|-tangential, of least possible positive degree. In particular it is irreducible and the map |$B(T) \mapsto B(P_t(T)) $| is an injective homomorphism from |$\mathbb{K}[T]$| into |$\theta_{0}^\infty(X,z)$|⁠. We check its surjectivity as follows, dividing any other |$0$|-tangential polynomial |$M(T)$| by |$P_t(T),$| we would get \[ M(T)=P_t(T)Q(T)+R(T)\quad{\rm and}\quad M\left(T\,\textrm{-}\,\frac{1}{z}\right)=P_t\left(T\,\textrm{-}\,\frac{1}{z}\right)Q\left(T\,\textrm{-}\,\frac{1}{z}\right)+R\left(T\,\textrm{-}\,\frac{1}{z}\right) \] with |${\rm{deg}}R < {\rm{deg}} P_t$| and both |$M(T\,\textrm{-}\,\frac{1}{z})$| and |$P_t(T\,\textrm{-}\,\frac{1}{z})$| having all their coefficients holomorphic at |$q$|⁠. Hence |$Q(T\,\textrm{-}\,\frac{1}{z})$| and |$R(T\,\textrm{-}\,\frac{1}{z})$| also satisfy the latter condition. It follows that they are |$0$|-tangential, and |$R$| a constant by the minimality of |$m=m_q$|⁠. Assuming by induction that |$Q(T)$| is a polynomial in |$P_t(T)$| we conclude that it is also true for |$M(T)$|⁠. At last, let us suppose that |$P(T) \in \theta_{0}^\infty(X,z)$| decomposes as |$A(T)+f$| with |$A(T)\in \mathbb{K}[T]$| and |$A(0)=0$|⁠. For any |$a\in \mathbb{K}$| we immediately check that $$P(T+a)\in \theta_{0}^\infty(X,z)\quad \textrm{and} \quad P(T+a)\,\textrm{-}\,P(T)=A(T+a) \,\textrm{-}\,A(T)\in \mathbb{K}[T]\cap \theta_{0}^\infty(X,z)=\mathbb{K}.$$ Hence |$A(T+a) \,\textrm{-}\,A(T)=P(a)\,\textrm{-}\,P(0)=A(a)$|⁠. In other words, |$A(T)$| is an additive polynomial of degree |$p^s$| for some |$s \geq m_q$| and the local expansion of |$f$| at |$q$| must be equal to |$f(z)=A(\frac{1}{z})+O(1)$|⁠. Taking |$P_t(T):=A_t(T)+f_t$| to the power |$p^{s\textrm{-}m_q}$| we get another one of same degree |$p^{s}$|⁠. By an inductive argument we obtain an additive polynomial |$B \in \mathbb{K}[T]$| such that |$B(P_t)=P(T)$|⁠, hence up to a constant |$B(A_t)=A(T)$| and |$B(f_t)=f$|⁠. ■ Proposition 6.3. Let |$\Gamma_t \subset {{\mathcal S}}$| denote the irreducible curve associated to |$P_t(T):=A_t(T)+f_t $||$($|see |$4.1.4)$| and assume |$A_t(T)$| is a separable additive polynomial, that is, |$c_t:=\frac{\partial}{\partial_T}(A_t)\neq 0$|⁠. Then |$\Gamma_t$| is a smooth curve of genus |$(g\,\textrm{-}\,1)p^{m_q}+1$| and the projection |$\pi:={\pi_{{\mathcal S}}}_{|\Gamma_t}: \Gamma_t \to X$| is an étale, Abelian Galois |$0$|-tangential cover of degree |$p^{m_q}$|⁠, with Galois group the set of roots |$\{A_t(T)=0\} \subset \mathbb{K}$|⁠. □ Proof. The divisor |$\Gamma_t \subset {\mathcal S}$| is defined on |${{\mathcal S}}\setminus {{\mathcal S}}_q$| and in a neighborhood of the fiber |${{\mathcal S}}_q$|⁠, by the equations |$A_t(T)+f_t=0$| and |$A_t(T)+O(z)=0,$| respectively. Hence |$\Gamma_t $| is linearly equivalent to |${\boldsymbol{p}}^{m_q}C_o$| and has genus |$(g\,\textrm{-}\,1)p^{m_q}+1$| according to the adjunction formula. It is also invariant by the group of automorphisms |$\{\tau_a:{{\mathcal S}} \to {{\mathcal S}}, A_t(a)=0\}$| (cf., |$4.1.3$|⁠) which has cardinal |${\boldsymbol{p}}^{m_q}$| because |$A_t$| is separable. The latter properties imply that |$\Gamma_t$| is smooth and |$\pi:\Gamma_t \to X$| is an étale, Galois, |$0$|-tangential cover with Galois group isomorphic to |$\{A_t(T)=0\}$| as asserted. ■ Theorem 6.4. Let |$A(T)+f\in \theta_{0}^\infty(X,z)$| be a |$0$|-tangential polynomial with |$A\in \mathbb{K}[T]$| an additive polynomial of degree |$ \textbf{p}^{m}\geq \textbf{p}^{m_q}$| and |$c$| denote the constant |$c:= \frac{\partial}{\partial T}(A)\in \mathbb{K}$|⁠. We fix |$a\notin \{0,c\}$| and consider hereafter the divisor of zeros on |${\mathcal S}$| of |$P(T)=A(T)\,\textrm{-}\,aT+f$|⁠, say |$\Gamma$|⁠, and the degree-|$\textbf{p}^{m}$| projections |$\pi:{\pi_{{\mathcal S}}}_{|\Gamma} : \Gamma \to X$| and |$\kappa:={\kappa_{{\mathcal S}}}_{|\Gamma} : \Gamma \to \mathbb{P}^1$||$($|see |$4.1.1$| and |$4.2)$|⁠. They satisfy|$:$| |$P(T)$| is a separable irreducible |$1$|-tangential polynomial of degree |$ \textbf{p}^{m}\geq \textbf{p}^{m_q}$|⁠. |$\Gamma$| is a smooth irreducible curve of genus |$g\textbf{p}^{m}$| and |$\pi$| is wildly ramified at |$\{r\}:=\pi^{\textrm{-}1}(q);$| |$\pi$| is an Abelian Galois cover étale over |$X\setminus\{q\}$| with Galois group |$\{A(T)\,\textrm{-}\,aT=0\}\subset \mathbb{K};$| the marked cover |$\pi: (\Gamma,r) \to (X,q)$| is |$1$|-tangential and |$\kappa$| an associated tangential function|$;$| let |$\lambda$| denote the local coordinate at |$r\in \Gamma$| such that |$\kappa+\pi^\ast(\frac{1}{z})= \frac{1}{\lambda}$|⁠. Then |$a\kappa = A(\frac{1}{\lambda})+O(1)$| implying |$A(T)+a\kappa \in \theta_{0}^\infty(\Gamma,\lambda)$| and |$m_r:={\rm{dim}} \overline{Orb}_F(r) \leq m$|⁠. □ Proof. The polynomial |$A(T)+f$| being |$0$|-tangential means that |$f(z)=A(\frac{1}{z})+O(1)$|⁠. It immediately follows that $$ \frac{\partial}{\partial T}(P(T))=c\,\textrm{-}\,a\neq 0\quad\textrm{ and }\quad P\left(T\,\textrm{-}\,\frac{1}{z}\right) =A(T)\,\textrm{-}\,aT+\frac{a}{z}+O(1). $$ Hence |$(zP(T\,\textrm{-}\,\frac{1}{z}))_{|z=0}=a\neq 0$| and |$P(T)$| is a separable |$1$|-tangential polynomial. On the other hand, according to the irreducibility criterion |$3.3$| it can only factor as |$P(T)=R_0(T)R_1(T)$| with |$R_0(T) \in\theta_{0,n_0}(X,z)$| and |$R_1(T) \in\theta_{1,n_1}(X,z)$|⁠, for some |$n_0,n_1\in \mathbb{N}^\ast$| such that |$n_0+n_1={\boldsymbol{p}}^{m}$|⁠. Recall at this point that for any |$0$|-tangential polynomial |$R(T)$| all the coefficients of |$R(T\,\textrm{-}\,\frac{1}{z})$| are holomorphic at |$q$| and |$deg\big(R(T\,\textrm{-}\,\frac{1}{z})_{|z=0}\big)= degR(T)$|⁠. The factorization would then imply the equalities $$ a= \left(zP\left(T\,\textrm{-}\,\frac{1}{z}\right)\right)_{|z=0}=R_0 \left(T\,\textrm{-}\,\frac{1}{z}\right)_{|z=0}\left(zR_1 \left(T\,\textrm{-}\,\frac{1}{z}\right)\right)_{|z=0}, $$ hence |$ n_0={\rm{deg}}\big(R_0(T\,\textrm{-}\,\frac{1}{z})_{|z=0}\big)\leq 0$|⁠. Contradiction! The irreducible curve |$\Gamma \subset {\mathcal S}$| is defined on |${{\mathcal S}}\setminus {{\mathcal S}}_q$| and in a neighborhood of the fiber |${{\mathcal S}}_q$|⁠, by the equations |$A(T)\,\textrm{-}\,aT+f=0$| and |$zA(T)\,\textrm{-}\,azT+a+O(z)=0$|⁠, respectively. In particular |$\Gamma$| is linearly equivalent to |${\boldsymbol{p}}^{m}C_o +{{\mathcal S}}_q$|⁠, of genus |$g{\boldsymbol{p}}^{m}$| according to the adjunction formula (see |$4.2.1$|⁠) and smooth on |${{\mathcal S}}\setminus {{\mathcal S}}_q$| because |$\frac{\partial}{\partial T}(P(T))=c\,\textrm{-}\,a \neq 0$|⁠. It also intersects |${{\mathcal S}}_q$| at the unique point in |$C_o \cap {{\mathcal S}}_q$|⁠, denoted hereafter by |$r$|⁠, with $$ zT^{\,\textrm{-}\,{{\boldsymbol{p}}}^m} P \left(T\,\textrm{-}\,\frac{1}{z}\right)=z+ aT^{\,\textrm{-}\,{{\boldsymbol{p}}}^m}+(c\,\textrm{-}\,a)zT^{\,\textrm{-}({\boldsymbol{{p}}}^m\,\textrm{-}\,1)}+zT^{\,\textrm{-}\,{{\boldsymbol{p}}}^m}O(1) =0 $$ giving a local equation around |$r$|⁠. Hence, |$\Gamma$| is smooth and |$\pi$| wildly ramified at |$r$|⁠. The curve |$\Gamma$| is invariant by the group of automorphisms |$\{\tau_b:{{\mathcal S}} \to {{\mathcal S}}, A(b)\,\textrm{-}\,ab=0\}$| (cf., |$4.1.3)$|⁠, which has cardinal |${\boldsymbol{p}}^{m}$| because |$A(T)\,\textrm{-}\,aT$| is a separable additive polynomial. The latter properties imply that |$\pi:\Gamma \to X$| is étale over |$X\setminus\{q\}$| and a Galois cover with Abelian Galois group isomorphic to |$\{A(T)\,\textrm{-}\,aT=0\}$| as asserted. We immediately check that |$\kappa$| satisfies the Tangency criterion |$2.4$|⁠. By its very nature |$\kappa$| satisfies |$P(\kappa)=A(\kappa)\,\textrm{-}\,a\kappa+f=0$|⁠, while |$f= A(\frac{1}{z})+O(1)$|⁠, hence |$a\kappa=\frac{a}{\lambda}\,\textrm{-}\,\frac{a}{z}= A(\frac{1}{\lambda})+O(1)$|⁠. In particular |$A(T)+a\kappa$| is a |$0$|-tangential polynomial and |$m_r:={\rm{dim}} \overline{Orb}_F(r) \leq m$|⁠. ■ Remark 4. Starting from data |$(X,q,z,f,A)$| such that |$A\in \mathbb{K}[T]$| is additive and |$A(T)+f\in \theta_0^\infty(X,z)$|⁠, plus the choice of any |$a \notin \{0,\frac{\partial}{\partial T}(A)\}$|⁠, we have obtained a |$1$|-tangential abelian Galois cover |$\pi:(\Gamma,r)\to (X,q)$| equipped with an analogous data |$(\Gamma,r,\lambda,a\kappa,A)$|⁠. If |$g=1$| or, more generally, if |$q\in X$| is a so-called Cartier point (cf., [2, 2.1]), then |$m_q=1 $| and there exists |$A(T)+f \in\theta_0^\infty(X,z)$| with |$A(T)=T^{{\boldsymbol{p}}}+cT $| for some |$c \in \mathbb{K}$|⁠. In the latter case for any |$a \notin \{0,c\}$| the corresponding |$($|unique|$)$| point |$r\in \Gamma$| over |$q\in X$| is again a Cartier point, that is, |$m_r=1$|⁠. By iterating the above procedure we can associate to any sequence |$(a_n)_{\mathbb{N}^\ast}$| in |$\mathbb{K} \setminus\{0,\frac{\partial}{\partial T}(A)\}$| an infinite tower of Artin–Schreier type |$1$|-tangential covers as explained hereafter. □ Corollary 6.5 (infinite towers of Artin–Schreier type |$1$|-tangential covers). Fix any |$0$|-tangential polynomial |$A(T)+f\in \theta_0^\infty(X,z)$|⁠, with |$A(T) \in \mathbb{K}[T]$| an additive polynomial of degree |$\textbf{p}^m$||$(m \geq m_q)$| and let |$c$| denote the constant |$\frac{\partial}{\partial T}(A)$|⁠. Then, given any sequence |$(a_j)_{\mathbb{N}^\ast}$| in |$\mathbb{K} \setminus \{0,c\}$| there exists an infinite tower |$\{\pi_{j}:(\Gamma_{j},r_{j}) \to (\Gamma_{j\textrm{-}1},r_{j\textrm{-}1}),\;\; j\in \mathbb{N}^\ast\}$| of degree-|$\textbf{p}^m$||$1$|-tangential covers, equipped with corresponding tangential functions |$\{\kappa_j:\Gamma_j \to \mathbb{P}^1, \,j\in \mathbb{N}^\ast\}$|⁠, |$($|where |$(\Gamma_0,r_0)=(X,q))$|⁠, such that for each |$j\in \mathbb{N}^\ast:$| |$\Gamma_{j}$| is a smooth curve of genus |$g\textbf{p}^{jm};$| |$\pi_j$| is an abelian Galois cover with Galois group |$\{A(T)\,\textrm{-}\,a_jT=0\} \subset \mathbb{K},$| wildly ramified at |$r_j:= \pi_j^{\textrm{-}1}(r_{j\textrm{-}1})$| and étale everywhere else|$;$| |$\Gamma_{j} \setminus \{r_j\}$| is isomorphic to |$\{A(T)\,\textrm{-}\,a_jT+a_{j\textrm{-}1}\kappa_{j\textrm{-}1}=0\}$|⁠, where |$(a_0,\kappa_0):=(1,f_t);$| for any |$j \geq 2$| the composed cover |$\pi_1 \circ\cdots \circ \pi_j: (\Gamma_j,r_j) \to (X,q)$| is |$1$|-tangential and |$\kappa_j+ {\pi_j}^\ast(\kappa_{j\textrm{-}1})+\cdots+(\pi_2 \circ \cdots \circ \pi_j)^\ast(\kappa_1)$| its tangential function. Moreover, if |$q\in X$| is a Cartier point and we pick |$A(T)=T^{\textbf{p}}+cT$| as in Remark |$4.2$|⁠, we get a tower |$\{\pi_{j}:(\Gamma_{j},r_{j}) \to (\Gamma_{j\textrm{-}1},r_{j\textrm{-}1})\}$| of Artin–Schreier |$1$|-tangential covers marked at Cartier points. □ Appendix. Matrix KP Rational and Trigonometric Solitons The connections between |$d$|-tangential covers and polynomials remain valid in the flat set up over a singular curve |$X$|⁠. We work out hereafter the simplest case: a singular plane cubic. Communicated by Prof. Igor Krichever References [1] Babelon, O. , Billey, E. Krichever I. M. and Talon. M. “Spin generalization of the Calogero-Moser system and the Matrix |$KP$| equation.” Amer. Math. Soc. Transl. Ser. 2 no. 170 ( 1995 ): 83 – 119 . [2] Baker, M. “Cartier points on curves.” Int. Math. Res. Not. 7 ( 2000 ): 353 – 70 . Google Scholar Crossref Search ADS [3] Dubrovin, B. A. “Theta functions and non-linear equations.” Russian Math. Surveys 36 ( 1981 ): 11 – 80 . Google Scholar Crossref Search ADS [4] Krichever, I. M. “Integration of non-linear equations by the method of algebraic geometry.” Funct. Anal. Appl. 11 , no. 1 ( 1977 ): 15 – 31 . Google Scholar Crossref Search ADS [5] Krichever, I. M. “Elliptic solutions of the KP equation and integrable systems of particles.” Funct. Anal. Appl. 14 , no. 4 ( 1980 ): 45 – 54 . [6] Treibich, A. “Tangential polynomials and Elliptic Solitons.” Duke Math. J. 59 , no. 3 ( 1989 ): 611 – 27 . Google Scholar Crossref Search ADS [7] Treibich, A. “Matrix Elliptic Solitons.” Duke Math. J. 90 , no. 3 ( 1997 ): 523 – 47 . Google Scholar Crossref Search ADS [8] Treibich, A. “Tangential polynomials and matrix KdV elliptic solitons.” Funct. Anal. Appl. 50 , no. 4 ( 2016 ): 308 – 18 . Google Scholar Crossref Search ADS [9] Treibich, A. “Revêtements tangentiels et tours infinies d’Artin–Schreier.” C. R. Acad. Sci. 354 , no. 12 ( 2016 ): 1225 – 29 . Google Scholar Crossref Search ADS © The Author 2017. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Deng,, Yu;Germain,, Pierre

doi: 10.1093/imrn/rnx210pmid: N/A

Abstract We prove polynomial bounds on the |$H^s$| growth for the nonlinear Schrödinger equation set on a torus, in dimension 3, with super-cubic and sub-quintic nonlinearity. Due to improved Strichartz estimates, these bounds are better for irrational tori than they are for rational tori. 1 Introduction 1.1 Growth of Sobolev norms for nonlinear Schrödinger Consider the nonlinear Schrödinger (NLS) equation |$(i\partial_t+\Delta)u=|u|^{p-1}u$| set on a torus. In subcritical cases (⁠|$p<\frac{d+2}{d-2}$|⁠), and for the torus |$\mathbb{T}^d = \mathbb{R}^d / \mathbb{Z}^d$|⁠, the global existence of finite energy (⁠|$H^1$|⁠) solutions is known since the foundational work of Bourgain [1]. Furthermore, the equation propagates regularity: if the data are smoother, say in |$H^s$| for |$s>1$|⁠, then |$u(t) \in H^s$| for all |$t$|⁠. The next question is to understand the qualitative behavior of the solution, and first of all: how fast may the |$H^s$| norm grow? This question is related to the phenomenon of weak turbulence which is generally described as the solution transferring energy to higher and higher frequencies, causing the |$H^s$| norm to grow while the |$H^1$| norm remains bounded. The growth rate of |$H^s$| norm can be seen as a control of how fast this energy transfer is happening. In general, it is easy to obtain an exponential upper bound for the |$H^s$| norm, by iterating local in time theory. In [2], using his “high-low method,” Bourgain was first able to improve this to a bound that is polynomial in time, in the case of a cubic nonlinearity in two dimension and two dimension. Further improvements and extension to other dimensions and nonlinearities have since been made in [7, 14, 15, 17]; see also [8, 12, 13, 16, 19, 20] and the references therein for other dispersive models. The recent work of Bourgain and Demeter [5], which proves optimal Strichartz estimates for the linear Schrödinger equation on irrational tori, opened the door to new questions for the nonlinear problem. It also enabled the authors, together with Deng et al. [10], to show that irrational tori enjoy better Strichartz estimates on long time intervals than rational tori. The aim of this article is to show that these linear estimates can be used to obtain improved |$H^s$| growth bounds for NLS equations on irrational tori, compared with rational tori (see also the recent work [11], where the authors study the NLS equation on irrational tori from a different aspect). More precisely, we will consider the NLS equation, in dimension |$d=3$|⁠, with power |$3<p<5$|⁠. Remark 1.1. The range |$3<p<5$| seems optimal for our method, as when |$p\leq 3$| we do not get any improvement from irrationality (see (1.4)). The case |$p=5$| is much more involved and is treated in another paper by the first author [9]. (We do not consider the supercritical case |$p>5$| where even global well-posedness is unknown.) □ Remark 1.2. Our arguments also apply for any other dimension |$d\geq 4$|⁠, in which case we obtain an improvement in the irrational case compared to the rational case, provided |$2<p<(d+2)/(d-2)$|⁠. □ 1.2 Main results Consider a three-dimensional super-cubic sub-quintic, defocusing NLS equation, \begin{equation} (i\partial_t+\Delta)u=|u|^{p-1}u,\quad 3<p<5, \end{equation} (1.1) on |$\mathbb{R}\times\mathbb{T}_{\ell}^3$|⁠, where |$\mathbb{T}_{\ell}^3$| is a rectangular torus \[\mathbb{T}_\ell^3=[0,\ell_1]\times[0,\ell_2]\times[0,\ell_3],\quad \ell=(\ell_1,\ell_2,\ell_3).\] One can prove that (1.1) is globally well posed in |$H^1(\mathbb{T}_{\ell}^3)$| (see Proposition 3.1 below) with conserved energy \[E_{\ell}[u]= \int_{\mathbb{T}_{\ell}^3}\left(\frac{1}{2}|\nabla u|^2+\frac{1}{p+1}|u|^{p+1}\right)\,\mathrm{d}x.\] Now, consider a solution |$u$| to (1.1) such that |$u(0)\in H^s(\mathbb{T}_{\ell}^3)$| for some |$s>1$|⁠; by preservation of regularity one can show that |$u(t)\in H^s$| for all time (see Proposition 3.1). We are interested in controlling the possible growth of the |$H^s$| norm of |$u$|⁠. For simplicity, and to avoid some technical issues, in this paper we only consider the case |$s=2$|⁠. (The technical issue is due to the non-smoothness of |$|u|^{p-1}u$|⁠, which prevents us from choosing |$s$| very large. Our proof works for any |$s\in(1,2]$|⁠, and probably for some values of |$s>2$|⁠. Since the purpose of this paper is just to provide an example of improvement due to irrationality, we will not pursue this question much further.) We will prove the following: Theorem 1.3. Suppose |$u$| is a solution to (1.1) with energy |$E_{\ell}[u]=E$| and |$\|u(0)\|_{H^2(\mathbb{T}_{\ell}^3)}=A$|⁠. Then we have the following. (1) For any choice of |$\ell$|⁠, we have \begin{equation} \|u(t)\|_{H^2(\mathbb{T}_{\ell}^3)}\lesssim A+(1+|t|)^{\frac{2}{5-p}+\delta} \end{equation} (1.2) for any time |$t$|⁠, and any |$\delta>0$|⁠. Here and below all implicit constants will depend on |$\ell$|⁠, |$E$|⁠, |$p$|⁠, and |$\delta$|⁠, but not on |$A$| or |$t$|⁠. (2) For generic choice of |$\ell$|⁠, that is excluding a subset of |$(\mathbb{R}^+)^3$| with measure |$0$|⁠, we have \begin{equation} \|u(t)\|_{H^2(\mathbb{T}_{\ell}^3)}\lesssim A+(1+|t|)^{\frac{2}{5-p+\theta(p)}} \end{equation} (1.3) for any time |$t$|⁠, where \begin{equation} \theta(p)=\frac{\min(p-3,5-p)}{182}, \end{equation} (1.4) which is positive for |$3<p<5$|⁠. □ Remark 1.4. Our choice of |$\theta(p)$| is clearly far from optimal. The point here is that generic irrational tori enjoy strictly better estimates, in terms of growth of higher Sobolev norms of solutions to NLS equations, than rational ones. □ Remark 1.5. Theorem 1.3 is actually true assuming some weaker Diophantine condition for |$\ell$|⁠, for example when \[\left|\frac{n_1}{\ell_1^2}+\frac{n_2}{\ell_2^2}+\frac{n_3}{\ell_3^2}\right|\geq C^{-1}(|n_1|+|n_2|+|n_3|)^{-4}\] for integers |$n_1$|⁠, |$n_2$|⁠, |$n_3$|⁠, not all zero (for generic |$\ell$| the exponent |$4$| can be replaced by |$2+\delta$|⁠); in particular Theorem 1.3 is true for |$\ell=(1,\sqrt[4]{2},\sqrt[4]{3})$|⁠. □ 1.3 Idea of the proof We describe here the main idea of the article; for relevant notations and spaces see Sections 1.4 and 2.2. Bourgain’s original proof of polynomial growth uses a high–low decomposition; here we shall use a variation of this idea that is technically more convenient, namely the upside down |$I$|-method. This appears in the work of Sohinger [14, 15], and is a variant of the original |$I$|-method developed by Colliander et al. [6]. In order to prove an upper bound \[\|u(t)\|_{H^2}\lesssim\max(\|u(0)\|_{H^2},(1+|t|)^{1/\kappa}),\] it suffices to show that \begin{equation} \text{If}\,\|u(t)\|_{H^2}\lesssim N,\,\text{then}\,\|u(t')\|_{H^2}\lesssim N for t\leq t'\leq t+N^{\kappa}. \end{equation} (1.5) Fix a scale |$N$|⁠, define a multiplier |$\mathcal{D}$| (see (1.9) below) such that |$\mathcal{D}=1$| for frequencies |$\ll N$| and |$\mathcal{D}=N^{-1}|\nabla|$| for frequencies |$\gtrsim N$|⁠, then using conservation of energy, we see that |$\|u(t)\|_{H^2}\lesssim N$| if and only if |$E_{\ell}[\mathcal{D}u]\lesssim 1$|⁠. The idea of the |$I$|-method is then to control the increment of the energy |$E_{\ell}[\mathcal{D}u]$|⁠. By the structure of the equation (1.1), one has that (Strictly speaking, |$(\mathcal{D}u)^{p-1}$| should be a |$p-1$|-homogeneous function of |$\mathcal{D}u$|⁠, but these are just trivial differences.) \[E_{\ell}[\mathcal{D}u(t_2)]-E_{\ell}[\mathcal{D}u(t_1)]\sim\int_{t_1}^{t_2}\int_{\mathbb{T}_{\ell}^3}(\nabla\mathcal{D}u)^2(\mathcal{D}u)^{p-1}\,\mathrm{d}x\mathrm{d}t.\] Moreover, the integrand above vanishes (or asymptotically vanishes) if the frequencies involved in |$(\mathcal{D}u)^{p-1}$| are all |$\ll N$|⁠, since |$\mathcal{D}=1$| for frequencies |$\lesssim N$| and that |$E_{\ell}[u]$| is conserved for (1.1). Therefore, one actually has that \begin{equation} E_{\ell}[\mathcal{D}u(t_2)]-E_{\ell}[\mathcal{D}u(t_1)]\sim\int_{t_1}^{t_2}\int_{\mathbb{T}_{\ell}^3}(\nabla\mathcal{D}u)^2(P_{\gtrsim N}\mathcal{D}u)(\mathcal{D}u)^{p-2}\,\mathrm{d}x\mathrm{d}t. \end{equation} (1.6) Suppose |$E_{\ell}[\mathcal{D}u(t)]$| is bounded, then by local theory one can bound |$\mathcal{D}u$| in |$X^{1,1/2+}$| locally (see Section 1.4 for notations and (2.4) for the definition of |$X^{s,b}$| spaces). By Hölder one has that \[|E_{\ell}[\mathcal{D}u(t_2)]-E_{\ell}[\mathcal{D}u(t_1)]|\lesssim\|\nabla\mathcal{D}u\|_{L_{t,x}^{10/3-}}^2\|P_{\gtrsim N}\mathcal{D}u\|_{L_{t,x}^{10/(6-p)+}}\|\mathcal{D}u\|_{L_{t,x}^{10-}}^{p-2},\] and by the |$X^{s,b}$| versions of Strichartz estimates (which follows from the usual Strichartz estimates of [5] and standard arguments in [18]) one has \[ \|\nabla\mathcal{D}u\|_{L_{t,x}^{10/3-}}\lesssim \|\mathcal{D}u\|_{X^{1,1/2+}},\,\, \|\mathcal{D}u\|_{L_{t,x}^{10-}}\lesssim \|\mathcal{D}u\|_{X^{1,1/2+}}; \,\, \|P_{\gtrsim N}\mathcal{D}u\|_{L_{t,x}^{10/(6-p)+}}\lesssim N^{(p-5)/2+}\|\mathcal{D}u\|_{X^{1,1/2+}} \] on time intervals of length |$1$|⁠, which implies that \begin{equation} \text{If}\,E_{\ell}[\mathcal{D}u(t_1)]\lesssim1,\,{\text{then}}\,|E_{\ell}[\mathcal{D}u(t_2)]-E_{\ell}[\mathcal{D}u(t_1)]|\lesssim N^{(p-5)/2+},\,{\text{when}}\,t_2-t_1\leq 1. \end{equation} (1.7) The gain |$N^{(p-5)/2}$| relies precisely on the subcritical nature of (1.1) when |$p<5$|⁠. By iteration, this then implies (1.5) for |$\kappa=(5-p)/2-$|⁠. The above is what happens for all tori; for (generic) irrational tori, one can resort to the long-time Strichartz estimate, which is proved in [10]: \[\|{\rm e}^{it\Delta}P_{\gtrsim N}\mathcal{D}f\|_{L_{t,x}^{10/(6-p)+}}\lesssim N^{(p-5)/2+}\|\mathcal{D}f\|_{H^1},\] on an interval of length |$N^{\gamma}$|⁠, where |$\gamma$| is sufficiently small. This in particular implies the bound \[\sum_{|m|\leq N^\gamma}\|{\rm e}^{it\Delta}P_{\gtrsim N}\mathcal{D}u(t_1)\|_{L_{t,x}^{10/(6-p)+}([m,m+1])}\lesssim N^{(p-5)/2+\gamma+}N^{-\gamma(6-p)/10}\cdot\|\mathcal{D}u(t_1)\|_{H^1}\] for the linear solution |${\rm e}^{it\Delta}\mathcal{D}u(t_1)$|⁠; one can then show that the same bound is true for the nonlinear solution |$\mathcal{D}u$| also, again thanks to the subcritical nature of (1.1). Plugging this into (1.6), we get that \begin{equation} \text{If}\,E_{\ell}[\mathcal{D}u(t_1)]\lesssim1,\,{\text{then}}\,|E_{\ell}[\mathcal{D}u(t_2)]-E_{\ell}[\mathcal{D}u(t_1)]|\lesssim N^{(p-5)/2+\gamma+}N^{-\gamma(6-p)/10},\,{\text{when}}\,t_2-t_1\leq N^{\gamma}, \end{equation} (1.8) in place of (1.7). Iterating (1.8) we get that (1.5) holds for \[\kappa=\gamma+\frac{5-p}{2}-\gamma+\frac{(6-p)\gamma-}{10}=\frac{5-p}{2}+\frac{(6-p)\gamma}{10}-,\] which improves upon the rational case. 1.4 Notations For a function |$f$| on |$\mathbb{R} \times \mathbb{T}^3$|⁠, and |$(\tau,k) \in \mathbb{R} \times \mathbb{Z}^3$|⁠, let $$\widehat{f}(\tau,k) = \int_{\mathbb{R} \times \mathbb{T}^3} f(t,x) {\rm e}^{-2\pi i (\tau t + k \cdot x)}\,\mathrm{d}x\,\mathrm{d}t.$$ For |$N$| a dyadic number, let |$P_N$|⁠, |$P_{<N}$| be the standard Littlewood–Paley projections; moreover for any set |$B\subset\mathbb{R}^3$| which is a ball, an annulus, a cube or a rectangular cuboid, we shall define the projection |$P_B$| in a similar way as |$P_N$|⁠. For any |$r\in\mathbb{R}$|⁠, we say a function |$F(z):\mathbb{C}\to\mathbb{C}$| is of type |$r$|⁠, if \[\left|\partial_z^m\partial_{\overline{z}}^nF\right|\lesssim_{m,n} |z|^{r-m-n}\] for all |$m,n\geq 0$|⁠; we denote by |$F_r$| a general function of type |$r$|⁠. For example, |$|z|^rz^m(\overline{z})^n$| is of type |$r+m+n$| for any |$r\in\mathbb{R}$| and |$m,n\in\mathbb{Z}$|⁠. For any fixed scale |$N$|⁠, define the multiplier |$\mathcal{D}$| to be \begin{equation} \widehat{\mathcal{D}u}(k)=m(k)\widehat{u}(k),\quad m(k)=\eta(k/N), \end{equation} (1.9) where |$\eta=\eta(y)$| is a smooth even function such that |$\eta(y)=1$| for |$|y|\leq 1$| and |$\eta(y)=|y|$| for |$|y|\geq 2$|⁠. We will use |$\chi$| to denote general cutoff functions: compactly supported, and equal to one in a neighborhood of zero. We write |$O(1)$| for a constant, and |$A \lesssim B$| if there exists a constant |$C$| such that |$A \leq CB$|⁠. We will use |$o(1)$| to denote any quantity that can be chosen arbitrarily small, and denote by |$a+$| (or |$a-$|⁠) anything larger (or smaller) than |$a$| that is |$o(1)$| close to |$a$|⁠. 2 Preparations 2.1 Change of variables First note that, by a change of variables, one can reduce (1.1) to the equation \begin{equation} (i\partial_t+\Delta_{\beta})u=|u|^{p-1}u, \end{equation} (2.1) on |$\mathbb{R}\times \mathbb{T}^3$|⁠, where |$\mathbb{T}^3=[0,1]^3$| is the standard square torus, and |$\Delta_{\beta}$| is the “anisotropic” Laplacian \begin{equation}\Delta_{\beta}=\beta_1\partial_{x_1}^2+\beta_2\partial_{x_2}^2+\beta_3\partial_{x_3}^2,\quad\beta_i=\ell_i^{-2}. \end{equation} (2.2) The mapping from |$\ell$| to |$\beta=(\beta_i)$| preserves zero measure sets, thus preserves genericity. The corresponding conserved energy for (2.1) is \begin{equation}\qquad E_{\beta}[u]=\int_{\mathbb{T}^3}\left(\frac{1}{2}\sum_{i=1}^3\beta_i|\partial_i u|^2+\frac{1}{p+1}|u|^{p+1}\right)\,\mathrm{d}x, \end{equation} (2.3) which, for simplicity, will be written as |$E[u]$| from now on. 2.2 Linear estimates Recall the definition of |$X^{s,b}$| and (for an interval |$I$| of |$\mathbb{R}$|⁠) |$X^{s,b,I}$| spaces \begin{gather} \|u\|_{X^{s,b}}^2=\sum_{k\in\mathbb{Z}^3}\int_{\mathbb{R}}\langle k\rangle^{2s}\langle\tau+ 2\pi Q(k)\rangle^{2b}|\widehat{u}(k,\tau)|^2\,\mathrm{d}\tau,\\ \|u\|_{X^{s,b,I}}=\inf_{g:\,g\equiv f\,\text{on}\,I}\|g\|_{X^{s,b}}, \end{gather} (2.4) where |$Q(k)=\beta_1k_1^2+\beta_2k_2^2+\beta_3k_3^2$|⁠. We have the following linear estimates. Proposition 2.1. Let |$\chi$| be a smooth cut-off. Parts (1)|$\sim$|(5) below hold for all |$\beta=(\beta_i)$|⁠, and part (6) holds for generic |$\beta_i$|⁠. (1) For all |$s,b\in\mathbb{R}$|⁠, one has \begin{equation}\|\chi(t)u\|_{X^{s,b}}\lesssim\|u\|_{X^{s,b}},\quad \|\chi(t){\rm e}^{it\Delta_{\beta}}f\|_{X^{s,b}}\lesssim\|f\|_{H^s}; \end{equation} (2.6) (2) For |$\varepsilon<1$| and |$-1/2<b\leq b'<1/2$|⁠, one has \begin{equation}\|\chi(\varepsilon^{-1}t)u\|_{X^{s,b}}\lesssim\varepsilon^{b'-b}\|u\|_{X^{s,b'}}; \end{equation} (2.7) (3) For |$1/2<b<1$|⁠, one has \begin{equation}\left\|\chi(t)\int_0^t {\rm e}^{i(t-t')\Delta_{\beta}}(u(t'))\,\mathrm{d}t'\right\|_{X^{s,b}}\lesssim\|u\|_{X^{s,b-1}}; \end{equation} (2.8) (4) For |$q\geq 10/3$| and |$b>1/2$|⁠, one has that \begin{equation}\|P_{N}u\|_{L_{t,x}^q([0,1]\times\mathbb{T}^3)}\lesssim N^{\frac{3}{2}-\frac{5}{q}+}\|P_Nu\|_{X^{0,b}}. \end{equation} (2.9) Moreover, the same bound holds if one replaces |$P_Nu$| by |$P_{B}u$|⁠, where |$B\subset\mathbb{R}^3$| any cube of size |$N$|⁠. (5) For |$q\geq 10/3$| and |$b>1/2$|⁠, one has that \begin{equation}\|P_{R}u\|_{L_{t,x}^q([0,1]\times\mathbb{T}^3)}\lesssim N^{\frac{3}{2}-\frac{5}{q}+}\left(\frac{M}{N}\right)^{\frac{1}{2}-\frac{5}{3q}}\|P_Ru\|_{X^{0,b}}, \end{equation} (2.10) where |$R\subset\mathbb{R}^3$| is any rectangular cuboid of dimensions |$N\times N\times M$| with |$M\leq N$|⁠. (6) For generic |$\beta=(\beta_i)$|⁠, and |$q>10/3$| and |$b>1/2$|⁠, one has that \begin{equation}\|P_{N}u\|_{L_{t,x}^q([0,N^{\nu(q)}]\times\mathbb{T}^3)}\lesssim N^{\frac{3}{2}-\frac{5}{q}+}\|P_Nu\|_{X^{0,b}}, \end{equation} (2.11) where \begin{align} \nu(q)=\left\{\begin{aligned} &\frac{4(3q-10)}{3q+14},&q<6,\\ &4,&q\geq 6.\end{aligned}\right.\notag\\[-1.8pc] \end{align} □ Proof. Parts (1)|$\sim$|(3) are well known, see [18]; part (4) follows from the full Strichartz estimate of Bourgain and Demeter [5]. The corresponding result for |$P_Bu$| follows from Galilean invariance. Part (6) is proved in [10]. Finally, part (5) follows from part (4) in the case |$q=10/3$|⁠, from Hausdorff–Young and Hölder in the case |$q=\infty$|⁠, and from interpolation for any |$q$| in between. ■ Remark 2.2. By interpolating (4) with the trivial bounds \[\|P_Nu\|_{L_{t,x}^2}=\|P_Nu\|_{X^{0,0}},\qquad \|P_{N}u\|_{L_{t,x}^{\infty}}\lesssim N^{\frac{3}{2}}\|P_Nu\|_{X^{0,1/2+}}\] and by duality, one gets a number of Strichartz estimates that will be used below; for example \[\|P_{N}u\|_{L_{t,x}^{10/3-}([0,1]\times\mathbb{T}^3)}\lesssim N^{o(1)}\|P_Nu\|_{X^{0,1/2-}}\] follows from interpolating the corresponding |$X^{0,1/2+}$| and |$X^{0,0}$| estimates. Moreover, by summing over |$N$| one also gets estimates such as \begin{align*} & \|u\|_{L_{t,x}^{10-}([0,1]\times\mathbb{T}^3)}\lesssim\|u\|_{X^{1,1/2+}}. \end{align*} □ 2.3 A nonlinear lemma For nonlinear terms of the type |$F(u)$|⁠, with |$F$| a function of type |$r$|⁠, one cannot employ the paraproduct decomposition to obtain estimates on dyadic frequency blocks. The following lemma circumvents this difficulty. Proposition 2.3 (A nonlinear lemma). Let |$F$| be any function of type |$r$|⁠, where |$r\geq 1$|⁠, then for any dyadic |$K$| we have \begin{equation}\|P_KF(u)\|_{L_{t,x}^{q_0}}\lesssim K^{o(1)}\|u\|_{L_{t,x}^{q_1}}^{r-1}\cdot\sum_{M}\min(1,K^{-1}M)\|P_Mu\|_{L_{t,x}^{q_2}} \end{equation} (2.12) provided that \[q_0,q_1,q_2\in [1,\infty],\quad \frac{1}{q_0}=\frac{r-1}{q_1}+\frac{1}{q_2}. \] □ Proof. We decompose \[P_KF(u)=\sum_{M}P_K\left[F(P_{\leq M}u)-F(P_{\leq M/2}u)\right]:=\sum_M P_K\mathcal{I}_M,\] and notice that \[\mathcal{I}_M=P_{M}u\cdot \int_0^1(\partial_zF)(P_{\leq M/2}u+\zeta P_{M}u)\,\mathrm{d}\zeta+\overline{P_{M }u}\cdot \int_0^1(\partial_{\overline{z}}F)(P_{\leq M/2}u+\zeta P_{M}u)\,\mathrm{d}\zeta.\] Therefore, when |$M\geq K$| we can estimate \[\|P_K\mathcal{I}_M\|_{L_{t,x}^{q_0}}\lesssim\|P_{M}u\|_{L_{t,x}^{q_2}}\left(\|P_{\leq M/2}u\|_{L_{t,x}^{q_1}}+\|P_{M}u\|_{L_{t,x}^{q_1}}\right)^{r-1}\lesssim \|P_{M}u\|_{L_{t,x}^{q_2}}\|u\|_{L_{t,x}^{q_1}}^{r-1},\] while for |$M\leq K$| we have \[\begin{split}\|P_K\mathcal{I}_M\|_{L_{t,x}^{q_0}}&\lesssim K^{-1}\left\|\nabla \left[F(P_{\leq M}u)-F(P_{\leq M/2}u)\right]\right\|_{L_{t,x}^{q_0}}\\ &\lesssim K^{-1}\left(\left\||\nabla P_{\leq M}u|\cdot|P_{\leq M}u|^{r-1}\right\|_{L_{t,x}^{q_0}}+\left\||\nabla P_{\leq M/2}u|\cdot|P_{\leq M/2}u|^{r-1}\right\|_{L_{t,x}^{q_0}}\right)\\ &\lesssim K^{-1}\|\nabla P_{\leq M}u\|_{L_{t,x}^{q_2}}\|P_{\leq M}u\|_{L_{t,x}^{q_1}}^{r-1}\\ &\lesssim K^{-1}\|u\|_{L_{t,x}^{q_1}}^{r-1}\sum_{M'\leq M}M'\|P_{M'}u\|_{L_{t,x}^{q_2}}.\end{split}\] Summing over |$M$|⁠, this implies (2.12). ■ Proposition 2.4. Recall the multiplier |$\mathcal{D}$| defined in (1.9). We have \begin{equation}\|\mathcal{D}(fg)\|_{L_{t,x}^{q_0}}\lesssim \|\mathcal{D}f\|_{L_{t,x}^{q_1}}\|\mathcal{D}g\|_{L_{t,x}^{q_2}},\end{equation} (2.13) provided \[ q_0,q_1,q_2\in[1,\infty],\quad \frac{1}{q_0}=\frac{1}{q_1}+\frac{1}{q_2}, \] and \begin{equation}\|\mathcal{D}F(u)\|_{L_{t,x}^q}\lesssim\|\mathcal{D}u\|_{L_{t,x}^{qr}}^r \end{equation} (2.14) provided that |$F$| is of type |$r$|⁠, where |$\min(q,r)\geq 1$|⁠. Moreover, we have \begin{equation}\|P_K\mathcal{D}F(u)\|_{L_{t,x}^{q_0}}\lesssim K^{o(1)}\|\mathcal{D}u\|_{L_{t,x}^{q_1}}^{r-1}\cdot\sum_{M}\min(1,K^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{q_2}}, \end{equation} (2.15) provided that |$F$| is of type |$r\geq 2$|⁠, and \[ q_0,q_1,q_2\in [1,\infty],\quad \frac{1}{q_0}=\frac{r-1}{q_1}+\frac{1}{q_2}. \] □ Proof. Note that \[\|\mathcal{D}u\|_{L_{t,x}^{p}}\sim\|u\|_{L_{t,x}^p}+N^{-1}\|\nabla u\|_{L_{t,x}^p},\] from which (2.13) and (2.14) follow easily. As for (2.15), we repeat the proof of Proposition 2.3, and write \[P_K\mathcal{D}F(u)=\sum_{M}P_K\mathcal{DI}_M,\] where \[\mathcal{I}_M=F(P_{\leq M}u)-F(P_{\leq M/2}u).\] By the same arguments in the proof of Proposition 2.3, together with (2.13) and (2.14), and using the fact that |$\nabla F$| is or type |$r-1\geq 1$|⁠, one gets that \[\|P_K\mathcal{DI}_M\|_{L_{t,x}^{q_0}}\lesssim \|\mathcal{D}u\|_{L_{t,x}^{q_1}}^{r-1}\|P_{M}\mathcal{D}u\|_{L_{t,x}^{q_2}}\] if |$M\geq K$|⁠, and that \[\|P_K\mathcal{DI}_M\|_{L_{t,x}^{q_0}}\lesssim K^{-1}\|\mathcal{D}u\|_{L_{t,x}^{q_1}}^{r-1}\sum_{M'\leq M}M'\|P_{M'}\mathcal{D}u\|_{L_{t,x}^{q_2}}\] when |$M\leq K$|⁠. Summing over |$M$| gives (2.15). ■ 3 Local Theory Proposition 3.1. Fix |$b=1/2+$|⁠. (1) (Local well-posedness) Suppose |$\|f\|_{H^1}\leq E$|⁠, then for a short time |$\varepsilon=\varepsilon(E)\ll 1$|⁠, the equation (2.1) has a unique solution |$u\in X^{1,b,[-\varepsilon,\varepsilon]}$| with initial data |$u(0)=f$|⁠, and one has \[\|u\|_{X^{1,b,[-\varepsilon,\varepsilon]}}\lesssim_E1.\] (2) (Propagation of regularity) Moreover, if in addition |$\|f\|_{H^2}\leq A$|⁠, then we also have \[ \|u\|_{X^{2,b,[-\varepsilon,\varepsilon]}}\lesssim_EA. \] □ Proof. This kind of proof is by now standard, see [1]; we include it here for the sake of completeness. (1) Fix |$b'=b+$|⁠. For a suitable cutoff function |$\chi$|⁠, we only need to prove that the mapping \[u\mapsto\mathcal{N}u:=\chi(t){\rm e}^{it\Delta_\beta}f-i\chi(t)\int_0^t{\rm e}^{i(t-t')\Delta_\beta}\left(\chi(\varepsilon^{-1}t')|u(t')|^{p-1}u(t')\right)\,\mathrm{d}t'\] is a contraction mapping from the set \[K:=\left\{u\in X^{1,b}:\|u-\chi(t){\rm e}^{it\Delta_\beta}f\|_{X^{1,b}}\leq 1\right\}\] to itself, since this would imply that the Duhamel map |$\mathcal{N}$| is also a contraction mapping from the |$1$|-neighborhood of |${\rm e}^{it\Delta_\beta}f$| in |$X^{1,b,[-\varepsilon,\varepsilon]}$| to itself. Now suppose |$u\in K$|⁠, then \[\|u\|_{X^{1,b}}\lesssim E\] by (2.6), and by (2.6)|$\sim$|(2.8) together with the definition of |$\mathcal{N}$|⁠, we get that \begin{equation}\left\|\mathcal{N}u-\chi(t){\rm e}^{it\Delta_\beta}f\right\|_{X^{1,b}}\lesssim \varepsilon^{b'-b}\left\|\chi(t)|u|^{p-1}u\right\|_{X^{1,b'-1}}. \end{equation} (3.1) Thus, if we can prove that \begin{equation}\left\|\chi(t)\nabla(|u|^{p-1}u)\right\|_{X^{0,b'-1}}\lesssim E^p, \end{equation} (3.2) then (3.1) would imply that \[\left\|\mathcal{N}u-\chi(t){\rm e}^{it\Delta_\beta}f\right\|_{X^{1,b}}\lesssim\varepsilon^{b'-b}E^{p},\] which gives that |$\mathcal{N}u\in K$| when |$\varepsilon$| is small enough (strictly speaking one has to bound the |$X^{0,b'-1}$| norm of |$\chi(t)|u|^{p-1}u$| also, but that easily follows from the proof below). Now let us prove (3.2). Note that \[\nabla(|u|^{p-1}u)=F_{p-1}(u)\nabla u+F_{p-1}(u)\nabla\overline{u},\] where |$F_{p-1}(u)$| denotes a function of type |$p-1$|⁠; we will only prove the bound for |$J:=F_{p-1}(u)\nabla \overline{u}$|⁠, since the other term is similar. Write |$I:=F_{p-1}(u)$|⁠. Let |$N_2$| and |$N_3$| be dyadic scales and |$\max (N_2,N_3)=N$|⁠, we decompose \[J=\sum_{N_2,N_3}P_{N_2}I\cdot P_{N_3}\nabla\overline{u}.\] By Proposition 2.3 and Strichartz, we know that \begin{align} \|P_{N_2}I\|_{L_{t,x}^{(5/2)+}}&\lesssim N_2^{o(1)}\|u\|_{L_{t,x}^{10-}}^{p-2}\sum_{M}\min(1,N_2^{-1}M)\|P_Mu\|_{L_{t,x}^{[10/(6-p)]+}}\notag\\ &\lesssim N_2^{o(1)}E^{p-2}\sum_{M}\min(1,N_2^{-1}M)M^{\frac{p-5}{2}+}E\lesssim N_2^{\frac{p-5}{2}+}E^{p-1}. \end{align} (3.3) Thus if |$N_2\geq N^{1/2}$|⁠, by Strichartz and dual Strichartz we have \begin{align} \|\chi(t)P_{N_2}I\cdot P_{N_3}\nabla\overline{u}\|_{X^{0,b'-1}}&\lesssim N^{o(1)}\|P_{N_2}I\cdot P_{N_3}\nabla\overline{u}\|_{L_{t,x}^{(10/7)+}}\notag\\ &\lesssim N^{o(1)} \|P_{N_2}I\|_{L_{t,x}^{(5/2)+}}\|\nabla P_{N_3}u\|_{L_{t,x}^{10/3}}\lesssim N^{\frac{p-5}{4}+} E^{p}, \end{align} (3.4) which gives an acceptable contribution to obtain (3.2) since |$p-5<0$|⁠. Now assume |$N_2\leq N^{1/2}$|⁠, then |$N=N_3$|⁠. For fixed |$N_3$|⁠, we decompose \[P_{N_3}u=\sum_{B\in \mathcal{Q}_{N_2,N_3}}P_{B}u,\] where |$\mathcal{Q}_{N_2,N_3}$| is a partition of |$\{k\in\mathbb{R}^3:|k|\sim N_3\}$| by cubes of size |$N_2$|⁠. Therefore we have \[J=\sum_{N_2}\sum_{N_3}\sum_{B\in \mathcal{Q}_{N_2,N_3}}P_{N_2}I\cdot P_B\nabla\overline{u}.\] Now if |$B\in \mathcal{Q}_{N_2,N_3}$| and |$B'\in\mathcal{Q}_{N_2,N_3'}$|⁠, and either |$N_3\gg N_3'$| or |$N_3'\gg N_3$| or |$\mathrm{dist}(B,B')\gg N_3$|⁠, it is easily seen that the terms |$P_{N_2}I\cdot P_B\nabla\overline{u}$| and |$P_{N_2}I\cdot P_{B'} \nabla \overline{u}$| must have disjoint Fourier support, and are thus orthogonal. Therefore we have \[\|\chi(t)J\|_{X^{0,b'-1}}\lesssim\sum_{N_2}\bigg(\sum_{N_3}\sum_{B\in \mathcal{Q}_{N_2,N_3}}\|\chi(t)P_{N_2}I\cdot P_B\nabla\overline{u}\|_{X^{0,b'-1}}^2\bigg)^{1/2}.\] Moreover for |$B \in \mathcal{Q}_{N_2,N_3}$| we actually have \[P_{N_2}I\cdot P_{B}\nabla\overline{u}=P_{10B}(P_{N_2}I\cdot P_{B}\nabla\overline{u}),\] thus by (3.3), Strichartz, dual Strichartz, and Hölder we have \begin{multline}\|\chi(t)P_{N_2}I\cdot P_{B}\nabla\overline{u}\|_{X^{0,b'-1}}\lesssim N_2^{o(1)} \|P_{10B}(P_{N_2}I\cdot P_{B}\nabla\overline{u})\|_{L_{t,x}^{(10/7)+}}\\\lesssim N_2^{o(1)} \|P_{N_2}I\|_{L_{t,x}^{(5/2)+}}\|\nabla P_{B}u\|_{L_{t,x}^{10/3}}\lesssim N_2^{\frac{p-5}{2}+}E^{p-1}\|P_{B}u\|_{X^{1,b}}, \end{multline} (3.5) which gives \[\sum_{N_3}\sum_{B\in\mathcal{Q}_{N_2,N_3}}\|\chi(t)P_{N_2}I\cdot P_B\nabla\overline{u}\|_{X^{0,b'-1}}^2\lesssim N_2^{(p-5)+}E^{2(p-1)}\|u\|_{X^{1,b}}^2.\] Taking square root and summing in |$N_2$|⁠, we get that \[\|\chi(t)J\|_{X^{0,b'-1}}\lesssim E^{p-1}\|u\|_{X^{1,b}}\lesssim E^{p},\] and this proves (3.2). To show that |$\mathcal{N}$| is a contraction mapping, it suffices to show that \[\left\|\chi(t)\nabla(|u|^{p-1}u-|v|^{p-1}v)\right\|_{X^{0,b'-1}}\lesssim E^{p-1}\|u-v\|_{X^{1,b}}\] provided |$\|u\|_{X^{1,b}}+\|v\|_{X^{1,b}}\lesssim E$|⁠. This can be done by writing \[\nabla(|u|^{p-1}u-|v|^{p-1}v)=\nabla\bigg((u-v)\cdot\int_0^1F_{p-1}(v+\zeta(u-v))\,\mathrm{d}\zeta +\overline{u-v}\cdot\int_0^1F_{p-1}(v+\zeta(u-v))\,\mathrm{d}\zeta\bigg). \] Fix |$\theta$| and let |$v+\theta(u-v)=w$|⁠, we only need to estimate the terms \[F_{p-1}(w)\nabla(u-v),\quad F_{p-1}(w)\nabla(\overline{u-v}),\quad (u-v)F_{p-2}(w)\nabla w,\quad \overline{u-v}\cdot F_{p-2}(w)\nabla w.\] The first two terms can be estimated in exactly the same way as above, and the next two terms can be estimated in the same way as the last one. Now let us prove the estimate for |$\overline{u-v}\cdot F_{p-2}(w)\nabla w$|⁠. Let |$I'=\overline{u-v}\cdot F_{p-2}(w)$|⁠, then if can prove \begin{equation}\|P_{N_2}I'\|_{L_{t,x}^{(5/2)+}}\lesssim N_2^{\frac{p-5}{2}+}E^{p-2}\|u-v\|_{X^{1,b}}, \end{equation} (3.6) the same argument as above will apply to give the desired estimate. To prove (3.6), we perform a similar (and simpler) argument as in the proof of Proposition 2.3. Write |$I'=I''+R'$|⁠, where |$I''=P_{\leq N_2}(\overline{u-v}) \cdot F_{p-2}(P_{\leq N_2}w)$|⁠, thus \[|\nabla I''|\lesssim|\nabla P_{\leq N_2}(u-v)|\cdot|P_{\leq N_2}w|^{p-2}+|P_{\leq N_2}(u-v)|\cdot|P_{\leq N_2}w|^{p-3}\cdot|\nabla P_{\leq N_2}w|\] and \[ |R''|\lesssim |P_{>N_2}(u-v)|\cdot|P_{\leq N_2}w|^{p-2}+|P_{>N_2}w|\cdot(|w|+|P_{\leq N_2}w|)^{p-3}|u-v|,\] (which follows from the inequality |$|F_{p-2}(x)-F_{p-2}(y)|\lesssim|x-y|\cdot(|x|^{p-3}+|y|^{p-3})$| since |$p>3$|⁠). Then we can bound \[ \begin{split} \|P_{N_2}I''\|_{L_{t,x}^{(5/2)+}}\lesssim N_2^{-1}\|\nabla P_{N_2}I''\|_{L_{t,x}^{(5/2)+}}&\lesssim N_2^{-1}\bigg(\|\nabla P_{\leq N_2}(u-v)\|_{L_{t,x}^{10/3}}\|P_{\leq N_2}w\|_{L_{t,x}^{10(p-2)}}^{p-2}\\ &\quad+\|\nabla P_{\leq N_2}w\|_{L_{t,x}^{10/3}}\|P_{\leq N_2}w\|_{L_{t,x}^{10(p-2)}}^{p-3}\|P_{\leq N_2}(u-v)\|_{L_{t,x}^{10(p-2)}}\bigg)\\ &\lesssim N_2^{\frac{p-5}{2}+}E^{p-1}\|u-v\|_{X^{1,b}} \end{split} \] and \begin{equation*} \begin{split} \|P_{N_2}R'\|_{L_{t,x}^{(5/2)+}}&\lesssim \|P_{>N_2}(u-v)\|_{L_{t,x}^{[10/(6-p)]+}}\|P_{\leq N_2}w\|_{L_{t,x}^{10-}}^{p-2}\\ &\quad+\|P_{>N_2}w\|_{L_{t,x}^{[10/(6-p)]+}}\left(\|w\|_{L_{t,x}^{10-}}+\|P_{\leq N_2}w\|_{L_{t,x}^{10-}}\right)^{p-3}\\ &\quad\times\left(\|u-v\|_{L_{t,x}^{10-}}+\|P_{\leq N_2}(u-v)\|_{L_{t,x}^{10-}}\right)\\ &\lesssim N_2^{\frac{p-5}{2}+}E^{p-1}\|u-v\|_{X^{1,b}},\end{split} \end{equation*} which completes the proof that |$\mathcal{N}$| is a contraction mapping. (2) Assume that |$\|f\|_{H^2}\leq A$|⁠. Using the same methods as above, we only need to show that |$\|u\|_{X^{2,b}}\lesssim A$| and |$\|u\|_{X^{1,b}}\lesssim E$| implies \[\left\|\chi(t)\nabla^2(|u|^{p-1}u)\right\|_{X^{0,b'-1}}\lesssim E^{p-1}A.\] Notice that \[\nabla^2(|u|^{p-1}u)=F_{p-1}(u)\nabla^2 u+F_{p-1}(u)\nabla^2\overline{u}+F_{p-2}(u)(\nabla u)^{2}+ F_{p-2}(u)(\nabla u)\cdot(\nabla\overline{u})+F_{p-2}(u)(\nabla\overline{u})^{2},\] where we use the convention that |$F_r$| denotes a function of type |$r$|⁠. The bounds for |$F_{p-1}(u)\nabla^2 u$| and |$F_{p-1}(u)\nabla^2\overline{u}$| is proved in exactly the same way as above, using |$\|u\|_{X^{2,b}}$| to control the |$\nabla^2u$| and |$\nabla^2\overline{u}$| factors; for the other terms we will only consider |$F_{p-2}(u)(\nabla u)\cdot(\nabla\overline{u})$|⁠, the rest being similar. Decompose \[F_{p-2}(u)(\nabla u)\cdot(\nabla\overline{u})=\sum_{N_2,N_3,N_4}P_{N_2}F_{p-2}(u)\cdot P_{N_3}\nabla u\cdot P_{N_4}\nabla\overline{u},\] and denote |$\max(N_2,N_3,N_4)=N$|⁠. Without loss of generality we may assume |$N_4\geq N_3$|⁠; if |$N_4\gtrsim N$| we have \[\begin{split}\mathcal{M}&:=\left\|\chi(t)P_{N_2}F_{p-2}(u)\cdot P_{N_3}\nabla u\cdot P_{N_4}\nabla\overline{u}\right\|_{X^{0,b'-1}}\\&\lesssim N^{o(1)} \left\|P_{N_2}F_{p-2}(u)\cdot P_{N_3}\nabla u\cdot P_{N_4}\nabla\overline{u}\right\|_{L_{t,x}^{10/7+}}\\ &\lesssim N^{o(1)} \|P_{N_2}F_{p-2}(u)\|_{L_{t,x}^{[10/(p-2)]-}}\|P_{N_3}\nabla u\|_{L_{t,x}^{[10/(6-p)]+}}\|P_{N_4}\nabla\overline{u}\|_{L_{t,x}^{10/3}}\\ &\lesssim N^{o(1)} \|u\|_{L_{t,x}^{10-}}^{p-2}\|P_{N_3}\nabla u\|_{L_{t,x}^{[10/(6-p)]+}}\|P_{N_4}\nabla\overline{u}\|_{L_{t,x}^{10/3}}\\ &\lesssim N^{o(1)} E^{p-2}N_3^{1+\frac{p-5}{2}+}E\cdot N_4^{-1+}A\lesssim N^{\frac{p-5}{2}+}E^{p-1}A,\end{split}\] and when |$N_4\ll N$| we must have |$N_2\gtrsim N$|⁠, so by Proposition 2.3 we have \begin{multline}\|P_{N_2}F_{p-2}(u)\|_{L_{t,x}^{10/3+}}\lesssim N_2^{o(1)}\|u\|_{L_{t,x}^{10-}}^{p-3}\sum_{M}\min(1,N_2^{-1}M)\|P_{M}u\|_{L_{t,x}^{[10/(6-p)]+}}\\ \lesssim N_2^{o(1)}E^{p-3}\sum_{M}\min(1,N_2^{-1}M)M^{\frac{p-5}{2}+}E\lesssim N_2^{\frac{p-5}{2}+}E^{p-2}, \end{multline} thus \[\begin{split}\mathcal{M}&\lesssim N^{o(1)} \left\|P_{N_2}F_{p-2}(u)\cdot P_{N_3}\nabla u\cdot P_{N_4}\nabla\overline{u}\right\|_{L_{t,x}^{10/7+}}\\ &\lesssim N^{o(1)} \|P_{N_2}F_{p-2}(u)\|_{L_{t,x}^{10/3+}}\|P_{N_3}\nabla u\|_{L_{t,x}^{10-}}\|P_{N_4}\nabla\overline{u}\|_{L_{t,x}^{10/3}}\\ &\lesssim N^{\frac{p-5}{2}+}E^{p-2}\cdot N_3N_{4}^{-1}\|P_{N_3}u\|_{L_{t,x}^{10-}}\|P_{N_4}\nabla^2\overline{u}\|_{L_{t,x}^{10/3}}\\ &\lesssim N^{\frac{p-5}{2}+}E^{p-1}A,\end{split}\] as desired. ■ 4 Proof of Theorem 1.3: General Case We shall use the I-method. Recall the multiplier |$\mathcal{D}$| defined in (1.9). Proposition 4.1. Suppose |$\|u(0)\|_{H^1} \leq E$| and |$\| \mathcal{D} u(0) \|_{H^1} \leq C_1 E$|⁠, for a constant |$C_1>0$|⁠. Choose |$\varepsilon$| as in Proposition 3.1. Then we have \[\big|E[\mathcal{D}u(T)]-E[\mathcal{D}u(0)]\big|\lesssim N^{\max(p-5,-1)+}+N^{o(1)}\sum_{M}M^{o(1)}\min(1,N^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}([0,\varepsilon]\times\mathbb{T}^3)} \] for |$0\leq T\leq \varepsilon$|⁠, with the energy functional |$E[u]$| defined in (2.3). □ Remark 4.2. In the above inequality, we chose to keep on the right-hand side the expression |$\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}([0,\varepsilon]\times\mathbb{T}^3)}$| instead of estimating it by |$M^{\frac{p-5}{2}}$|⁠, which would be possible through the |$X^{s,b}$| norm derived in the proof below. This will be crucial to get the improvement on irrational tori: indeed, we will prove in the next section that, on irrational tori, a better estimate of |$\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}$| becomes available. □ Proof. Step 1: the modified energy indentity. From the assumption, we know that (recall that implicit constants depend on |$E$|⁠) \[\|u(0)\|_{H^1}\lesssim 1,\quad \|u(0)\|_{H^2}\lesssim N.\] By Proposition 3.1, one has that \[\|u\|_{X^{1,b,[-\varepsilon,\varepsilon]}}\lesssim1,\quad \|u\|_{X^{2,b,[-\varepsilon,\varepsilon]}}\lesssim N.\] This gives that \[\|\mathcal{D}u\|_{X^{1,b,[-\varepsilon,\varepsilon]}}\lesssim1.\] By considering a suitable extension of |$u$|⁠, we may assume |$\|\mathcal{D}u\|_{X^{1,b}}\lesssim1$|⁠. Now let us compute the time evolution of |$E[\mathcal{D}u]$|⁠. In fact, one has that \[(i\partial_t+\Delta_{\beta})(\mathcal{D}u)=|\mathcal{D}u|^{p-1}(\mathcal{D}u)+\mathcal{R},\] where \[\mathcal{R}=\mathcal{D}(|u|^{p-1}u)-|\mathcal{D}u|^{p-1}(\mathcal{D}u).\] Now by conservation of energy for (2.1), one has that \begin{equation}\begin{aligned}\partial_tE[\mathcal{D}u]&=\int_{\mathbb{T}^3}\left\{\sum_{i=1}^3\beta_i\Re(\partial_i\overline{\mathcal{D}u}\cdot\partial_i(\mathcal{D}u)_t)+\Re(\overline{\mathcal{D}u}\cdot(\mathcal{D}u)_t)\cdot|\mathcal{D}u|^{p-1}\right\}\,\mathrm{d}x\\ &=\sum_{i=1}^3\beta_i\Im\int_{\mathbb{T}^3}(\partial_i\overline{\mathcal{D}u}\cdot \partial_i\mathcal{R})\,\mathrm{d}x+\Im\int_{\mathbb{T}^3}|\mathcal{D}u|^{p-1}\overline{\mathcal{D}u}\cdot\mathcal{R}\,\mathrm{d}x.\end{aligned} \end{equation} (4.1) Thus, upon integrating in |$t$|⁠, we reduce to estimating the space-time integrals \begin{equation}\int_{[0,T]\times\mathbb{T}^3}(\partial_i\overline{\mathcal{D}u}\cdot \partial_i\mathcal{R})\,\mathrm{d}x\mathrm{d}t \end{equation} (4.2) and \begin{equation}\int_{[0,T]\times\mathbb{T}^3}|\mathcal{D}u|^{p-1}\overline{\mathcal{D}u}\cdot\mathcal{R}\,\mathrm{d}x\mathrm{d}t. \end{equation} (4.3) Step 2: bound for (4.3). Let |$u_1=P_{\leq N/10}u$| and |$u_2=u-u_1=P_{>N/10}u$|⁠, note that \[\begin{aligned}\mathcal{R}&=\mathcal{D}(|u|^{p-1}u)-|\mathcal{D}u|^{p-1}(\mathcal{D}u)\\&=\mathcal{D}(|u|^{p-1}u-|u_1|^{p-1}u_1)-\left[|\mathcal{D}u|^{p-1}(\mathcal{D}u)-|\mathcal{D}u_1|^{p-1}(\mathcal{D}u_1)\right]-(1-\mathcal{D})\left(|u_1|^{p-1}u_1\right).\end{aligned}\] For the first term, using the identity \[|u|^{p-1}u-|u_1|^{p-1}u_1=u_2\cdot\int_0^1F_{p-1}(u_1+\zeta u_2)\,\mathrm{d}\zeta+\overline{u_2}\cdot\int_0^1F_{p-1}(u_1+\zeta u_2)\,\mathrm{d}\zeta,\] and using (2.13), (2.14), Hölder, and Strichartz, we can bound the corresponding contribution by \[ \begin{split} & \left\||\mathcal{D}u|^{p-1}\overline{\mathcal{D}u}\right\|_{L_{t,x}^{(10/p)-}}\cdot\left\|\mathcal{D}(|u|^{p-1}u-|u_1|^{p-1}u_1)\right\|_{L_{t,x}^{[10/(10-p)]+}}\\ &\quad \lesssim\|\mathcal{D}u\|_{L_{t,x}^{10-}}^{p}\|\mathcal{D}u_2\|_{L_{t,x}^{q_1}}\left(\|\mathcal{D}u\|_{L_{t,x}^{10-}}+\|\mathcal{D}u_1\|_{L_{t,x}^{10-}}\right)^{p-1}\lesssim N^{p-5}, \end{split} \] where |$q_1=10/(11-2p)+$|⁠. The same bound holds for the second term, using the fact that \[ \left||\mathcal{D}u|^{p-1}(\mathcal{D}u)-|\mathcal{D}u_1|^{p-1}(\mathcal{D}u_1)\right|\lesssim|\mathcal{D}u_2|\cdot(|\mathcal{D}u|+|\mathcal{D}u_1|)^{p-1}. \] For the last term, notice that \[ \begin{split} \left\|(1-\mathcal{D})\left(|u_1|^{p-1}u_1\right)\right\|_{L_{t,x}^{[10/(10-p)]+}} &\lesssim N^{-1}\|\nabla(|u_1|^{p-1}u_1)\|_{L_{t,x}^{[10/(10-p)]+}}\\ &\lesssim N^{-1}\|u_1\|_{L_{t,x}^{10-}}^{p-1}\|\nabla u_1\|_{L_{t,x}^{q_1}}\lesssim N^{-1}. \end{split} \] Gathering the estimates, we find that \[ \text{(4.3)}\lesssim N^{\max(p-5,-1)+}. \] Step 3: bound for (4.2). Note that \[\nabla\mathcal{R}=\mathcal{D}\left[F_{p-1}(u)\nabla u+F_{p-1}(u)\nabla\overline{u}\right]-\left[F_{p-1}(\mathcal{D}u)\nabla(\mathcal{D}u)+F_{p-1}(\mathcal{D}u)\nabla\overline{\mathcal{D}u}\right].\] Since the other term is similar, we only consider the term \[\mathcal{D}(F_{p-1}(u)\nabla u)-F_{p-1}(\mathcal{D}u)\nabla(\mathcal{D}u),\] which can be decomposed as \begin{equation}\mathcal{D}[(F_{p-1}(u)-F_{p-1}(u_1))\nabla u]-[F_{p-1}(\mathcal{D}u)-F_{p-1}(\mathcal{D}u_1)]\nabla(\mathcal{D}u)+[\mathcal{D}(\mathcal{H}\nabla u)-\mathcal{H}\nabla\mathcal{D}u], \end{equation} (4.4) where |$\mathcal{H}=F_{p-1}(u_1)$|⁠. For the first term in (4.4), denote |$\mathcal{L}=F_{p-1}(u)-F_{p-1}(u_1)$|⁠; note that \[\mathcal{L}=u_2\cdot\int_0^1 F_{p-2}(u_1+\zeta u_2)\,\mathrm{d}\zeta+\overline{u_2}\cdot\int_0^1F_{p-2}(u_1+\zeta u_2)\,\mathrm{d}\zeta.\] We see that \[\|\mathcal{D}P_K\mathcal{L}\|_{L_{t,x}^{5/2}}\lesssim \|\mathcal{D}u_2\|_{L_{t,x}^{[10/(6-p)]+}}\left(\|\mathcal{D}u_1\|_{L_{t,x}^{10-}}+\|\mathcal{D}u_2\|_{L_{t,x}^{10-}}\right)^{p-2}\lesssim \sum_{M\geq N/10}\|\mathcal{D}P_Mu\|_{L_{t,x}^{[10/(6-p)]+}}\] if |$K\leq N$|⁠, and (note that |$u_1=P_{\leq N/10}u$| satisfies the same estimates as |$u$|⁠) \begin{equation}\begin{split}\|\mathcal{D}P_K\mathcal{L}\|_{L_{t,x}^{5/2}}&\lesssim\left\|\mathcal{D}P_KF_{p-1}(u)\right\|_{L_{t,x}^{5/2}}+\left\|\mathcal{D}P_KF_{p-1}(u_1)\right\|_{L_{t,x}^{5/2}}\\&\lesssim K^{o(1)}\|\mathcal{D}u\|_{L_{t,x}^{10-}}^{p-2}\sum_{M}\min(1,K^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}\\&\lesssim K^{o(1)}\sum_{M}\min(1,K^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}\end{split} \end{equation} (4.5) by (2.15) if |$K\geq N$|⁠, which gives by summation in |$K$| that \begin{align} &\sum_K K^{o(1)}\|\mathcal{D}P_K\mathcal{L}\|_{L_{t,x}^{5/2}}\lesssim N^{o(1)}\notag\\ &\quad\times\bigg(\sum_{M\geq N/10}\|\mathcal{D}P_Mu\|_{L_{t,x}^{[10/(6-p)]+}}+\sum_{M}M^{o(1)}\min(1,N^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}\bigg). \end{align} (4.6) Then, the contribution corresponding to this term can be decomposed as \[\int_{[0,T]\times\mathbb{T}^3}\partial_i\overline{\mathcal{D}u}\cdot\mathcal{D}(\mathcal{L}\partial_iu)\,\mathrm{d}x\mathrm{d}t=\sum_{K}\sum_{B} \int_{[0,T]\times\mathbb{T}^3}\partial_iP_{10B}\overline{\mathcal{D}u}\cdot\mathcal{D}(P_K\mathcal{L}\cdot \partial_iP_Bu)\,\mathrm{d}x\mathrm{d}t,\] where for fixed |$K$|⁠, |$B$| runs over some partition of |$\mathbb{R}^3$| into cubes of size |$K$|⁠. By orthogonality, this is bounded by \begin{align*} &\sum_K\sum_B \left\|\partial_iP_{10B}\overline{\mathcal{D}u}\right\|_{L_{t,x}^{10/3}}\left\|\mathcal{D}P_K\mathcal{L}\right\|_{L_{t,x}^{5/2}}\left\|\mathcal{D}\partial_iP_Bu\right\|_{L_{t,x}^{10/3}}\\ &\quad\lesssim \sum_KK^{o(1)}\|\mathcal{D}P_K\mathcal{L}\|_{L_{t,x}^{5/2}}\bigg(\sum_B\left\|\partial_iP_{10B}\overline{\mathcal{D}u}\right\|_{X^{0,b}}^2\bigg)^{1/2}\bigg(\sum_B \left\|\mathcal{D}\partial_iP_Bu\right\|_{X^{0,b}}^{2}\bigg)^{1/2}\\ &\quad\lesssim N^{o(1)}\sum_{M}M^{o(1)}\min(1,N^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}. \end{align*} By analogous arguments, the same bound can be obtained for the second term in (4.4). For the last term in (4.4), we use the same trick and decompose \begin{align} &\int_{[0,T]\times\mathbb{T}^3}\partial_i\overline{\mathcal{D}u}\cdot[\mathcal{D}(\mathcal{H}\partial_iu)-\mathcal{H}\partial_i\mathcal{D}u]\,\mathrm{d}x\mathrm{d}t\notag\\ &\quad=\sum_{K}\sum_{B} \int_{[0,T]\times\mathbb{T}^3}\partial_iP_{10B}\overline{\mathcal{D}u}\cdot[\mathcal{D}(P_K\mathcal{H}\cdot \partial_iP_{B}u)-P_K\mathcal{H}\cdot \partial_i\mathcal{D}P_Bu]\,\mathrm{d}x\mathrm{d}t, \end{align} (4.7) where |$B$| runs over some partition of |$\mathbb{R}^3$| into cubes of size |$K$|⁠. By (2.15) and the fact that |$\|\mathcal{D}u\|_{L_{t,x}^{10-}}\lesssim 1$|⁠, we have \[\|\mathcal{D}P_K\mathcal{H}\|_{L_{t,x}^{5/2}}\lesssim K^{o(1)}\sum_{M\lesssim N}\min(1,K^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}.\] Moreover, by definition of |$\mathcal{D}$| we have \[\mathcal{F}[\mathcal{D}(P_K\mathcal{H}\cdot \partial_iP_{B}u)-P_K\mathcal{H}\cdot \partial_i\mathcal{D}P_Bu](k)=\sum_{l+m=k}\bigg[\eta\bigg(\frac{k}{N}\bigg)-\eta\bigg(\frac{m}{N}\bigg)\bigg]\widehat{P_K\mathcal{H}}(l)\widehat{\partial_iP_Bu}(m),\] and the symbol \[\eta\bigg(\frac{k}{N}\bigg)-\eta\bigg(\frac{m}{N}\bigg)=\frac{l}{N}\cdot\int_0^1\nabla \eta\bigg(\frac{m+\zeta l}{N}\bigg)\,\mathrm{d}\zeta,\] thus by Coifman–Meyer theory and transference principle we have \[\left\|\mathcal{D}(P_K\mathcal{H}\cdot \partial_iP_{B}u)-P_K\mathcal{H}\cdot \partial_i\mathcal{D}P_Bu\right\|_{L_{t,x}^{10/7}}\lesssim \min(1,KN^{-1})\|\mathcal{D}P_K\mathcal{H}\|_{L_{t,x}^{5/2}}\|\mathcal{D}\nabla P_Bu\|_{L_{t,x}^{10/3}}.\] After summing in |$K$| and |$B$| and using orthogonality, this gives that \begin{split}& \left|\int_{[0,T]\times\mathbb{T}^3}\partial_i\overline{\mathcal{D}u}\cdot[\mathcal{D}(\mathcal{H}\partial_iu)-\mathcal{H}\partial_i\mathcal{D}u]\,\mathrm{d}x\mathrm{d}t\right|\\ &\quad \lesssim\sum_K\sum_B \min(1,KN^{-1})\|\mathcal{D}P_K\mathcal{H}\|_{L_{t,x}^{5/2}}\| \mathcal{D} \nabla P_Bu\|_{L_{t,x}^{10/3}}\|\nabla P_{10B}\overline{\mathcal{D}u}\|_{L_{t,x}^{10/3}}\\ &\quad \lesssim\sum_{K}\min(1,KN^{-1})\| \mathcal{D} P_K\mathcal{H}\|_{L_{t,x}^{5/2}}\bigg(\sum_{B}\|\nabla P_B\mathcal{D}u\|_{X^{0,b}}^2\bigg)^{1/2}\bigg(\sum_{B}\|\nabla P_{10B}\overline{\mathcal{D}u}\|_{X^{0,b}}^2\bigg)^{1/2}\\ &\quad\lesssim\sum_{K}\min(1,KN^{-1})\cdot K^{o(1)}\sum_{M\lesssim N}\min(1,K^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}\\ &\quad\lesssim N^{o(1)}\sum_{M}M^{o(1)}\min(1,N^{-1}M)\|P_M\mathcal{D}u\|_{L_{t,x}^{[10/(6-p)]+}}.\end{split} (4.8) This completes the proof. ■ (Proof of Theorem 1.3 in the general case). First choose |$N = A = \| u(0) \|_{H^2}$| and observe that, by Sobolev embedding, $$E[ \mathcal{D} u(0) ] \leq C_0 E $$ for a constant |$C_0$|⁠. By Strichartz and Proposition 4.1, we know that $$ \| P_M \mathcal{D} u \|_{L_{t,x}^{[10/(6-p)]+}([0,\varepsilon]\times\mathbb{T}^3)}\lesssim M^{\frac{p-5}{2}+}. $$ As long as |$T$| is such that $$ \sup_{0\leq t\leq T}E[\mathcal{D}u(t)]\leq 2 C_0 E \qquad \forall t \in [0,T], $$ we learn from iterating Proposition 4.1 that $$ \sup_{0\leq t\leq T}E[\mathcal{D}u(t)] - C_0 E \lesssim N^{\max(p-5,-1)}T+N^{o(1)}\sum_MM^{o(1)}\min(1,N^{-1}M)\cdot TM^{\frac{p-5}{2}+}\lesssim N^{\frac{p-5}{2}+}T. $$ By a bootstrap argument, this gives that |$E[\mathcal{D}u(t)]\leq 2C_0 E$|⁠, and hence |$\|u(t)\|_{H^2}\leq 4C_0 A$|⁠, up to a time \[ T \sim A^{\frac{5-p}{2}-}. \] This implies immediately (1.2). ■ 5 Proof of Theorem 1.3: Irrational Case Let |$q_0=10/(6-p)+$| and |$\sigma=1/2-5/q_0=(p-5)/2+$| (notice that |$-1<\sigma<0$|⁠). By Proposition 3.1 and Strichartz, we know that if |$u$| is a solution to (2.1) with |$\|u(0)\|_{H^1}\leq E$|⁠, then \begin{equation} \|P_Ku\|_{L_{t,x}^{q_0}([0,\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+} \end{equation} (5.1) for any dyadic |$K$|⁠. The improvement in the irrational case will be based on an improvement of (5.1), namely we have the following Proposition 5.1. Suppose |$\beta=(\beta_i)$| satisfies the long-time Strichartz estimates (Proposition 2.1, part (6)). Under the assumption that |$u$| is a solution to (2.1) and \begin{equation} \sup_{0\leq t\leq \varepsilon K^{\gamma}}\|\mathcal{D}u(t)\|_{H^1}\lesssim 1, \end{equation} (5.2) we have for any |$\gamma>0$| that \begin{equation} \sum_{m=0}^{K^{\gamma}}\|P_K\mathcal{D}u\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\gamma+\sigma+}\cdot\max(K^{-\frac{\gamma}{q_0}},K^{\gamma+\theta_1}), \end{equation} (5.3) where \[ \theta_1=\frac{(p-3)(p-5)}{3(3p+7)}<0. \] □ Remark 5.2. The trivial bound obtain by iterating (5.1) would be |$K^{\gamma+\sigma+}$|⁠. We get an improvement for |$0<\gamma<|\theta_1|$|⁠. □ Proof. Step 1: decomposition of the nonlinear term. Fix |$b=1/2+$|⁠. The assumption (5.2) and Proposition 3.1 give a solution |$u$| such that |$\| u \|_{X^{1,b,[0,\epsilon]}} +\|\mathcal{D}u\|_{X^{1,b,[0,\varepsilon]}}\lesssim 1$|⁠. By considering a suitable extension we may assume |$\| u \|_{X^{1,b}} + \|\mathcal{D}u\|_{X^{1,b}}\lesssim 1$|⁠, and |$u$| is compactly supported in time. In this proof below we will fix two parameters |$\gamma_0>0$| and |$\gamma_1>4\gamma_0+$|⁠; they will be determined later. Let \[u_1=P_{\leq K/10}u,\quad u_2=u-u_1=P_{>K/10}u.\] By (2.1) and Taylor expansion one has that \begin{multline} (i\partial_t+\Delta_{\beta})P_K\mathcal{D}u=P_K\mathcal{D}(|u|^{p-1}u)=P_K\mathcal{D}\bigg\{|u_1|^{p-1}u_1+\frac{p+1}{2}|u_1|^{p-1}u_2+\overline{u_2}F_{p-1}(u_1)\\ +\int_0^1\left[F_{p-2}(u_1+\zeta u_2)u_2^2+F_{p-2}(u_1+\zeta u_2)u_2\overline{u_2}+F_{p-2}(u_1+\zeta u_2)(\overline{u_2})^2\right] \zeta\,\mathrm{d}\zeta\bigg\}. \end{multline} (5.4) Moreover, let |$K'=K^{\gamma_0}$| and define \[u_3=P_{\leq K'}u,\quad u_4=P_{(K',K/10]}u=u_1-u_3,\] then we can decompose further \[\frac{p+1}{2}|u_1|^{p-1}=h(t)+\mathbb{P}_{\neq 0}\Omega+u_4\cdot\int_0^1 F_{p-2}\left(u_3+\zeta u_4\right)\,\mathrm{d}\zeta+\overline{u_4}\cdot\int_0^1 F_{p-2}\left(u_3+\zeta u_4\right)\,\mathrm{d}\zeta,\] where \[h(t)=\mathbb{P}_0\Omega,\quad\Omega=\frac{p+1}{2}|u_3|^{p-1},\] and |$\mathbb{P}_0$| denotes the projection onto the zeroth mode. Therefore we get (notice that |$P_K\mathcal{D}u_2=P_K\mathcal{D}u$|⁠) \[(i\partial_t+\Delta_{\beta})P_K\mathcal{D}u=h(t) P_K\mathcal{D}u+ \mathcal{R}\,\] where $$ \mathcal{R} = \mathcal{R}_1 + \mathcal{R}_2 + \mathcal{R}_3 + \mathcal{R}_4 $$ and \begin{align*} \mathcal{R}_1&=P_K\mathcal{D}\left(|u_1|^{p-1}u_1\right),\\ \mathcal{R}_2&=P_K\mathcal{D}\left(\overline{u_2}\cdot F_{p-1}(u_1)\right),\\ \mathcal{R}_3&=P_K\mathcal{D}\left(u_2\cdot \mathbb{P}_{\neq 0}\Omega\right),\\ \mathcal{R}_4&=P_K\mathcal{D}\left\{u_2u_4\int_0^1F_{p-2}(u_3+\zeta u_4)\,\mathrm{d}\zeta+u_2\overline{u_4}\int_0^1F_{p-2}(u_3+\zeta u_4)\,\mathrm{d}\zeta\right\}\\ &\quad+P_K\mathcal{D}\left\{\int_0^1\left[F_{p-2}(u_1+\zeta u_2)u_2^2+2F_{p-2}(u_1+\zeta u_2)u_2\overline{u_2}+F_{p-2}(u_1+\zeta u_2)(\overline{u_2})^2 \right] \zeta\,\mathrm{d}\zeta\right\}\!. \end{align*} Let \[ v(t)=\exp\left(i\int_0^t h(t')\mathrm{d}t'\right)\cdot P_K\mathcal{D}u(t), \] then we have that \[(i\partial_{t}+\Delta_{\beta})v=\exp\left(i\int_0^t h(t')\mathrm{d}t'\right)\cdot\mathcal{R}:=\mathcal{R}',\] so \[v(t)=e^{it\Delta_{\beta}}v(0)-i\int_0^t e^{i(t-t')\Delta_{\beta}}\mathcal{R}'(t')\,\mathrm{d}t'\] for |$0\leq t\leq\varepsilon$|⁠, which gives that \[\|\chi(t)(v(t)-{\rm e}^{it\Delta_{\beta}}v(0))\|_{X^{1,b}}\lesssim \|\chi(t)\mathcal{R}'\|_{X^{1,b-1}}.\] We next proceed to estimate |$\mathcal{R}'$|⁠. We will denote |$\mathcal{R}_j'$| the term in |$\mathcal{R}'$| corresponding to |$\mathcal{R}_j$|⁠. Step 2: estimate of |$\mathcal{R}_1'$| and |$\mathcal{R}_3'$|⁠. First, we claim that \[\|\chi(t)\mathcal{R}_4'\|_{X^{1,b-1}}\lesssim K\|\mathcal{R}_4'\|_{L_{t,x}^{10/7+}}\lesssim K^{\sigma\gamma_0+}.\] In fact, we only need to consider |$\mathcal{R}_4$|⁠. For the term |$u_2u_4F_{p-2}(u_3+\theta u_4)$| (the other term being similar), one can bound \[K\left\|P_K\mathcal{D}\left(u_2u_4F_{p-2}(u_3+\theta u_4)\right)\right\|_{L_{t,x}^{10/7+}}\lesssim K^{o(1)}\cdot\|K\mathcal{D}u_2\|_{L_{t,x}^{10/3}}\|\mathcal{D}u_4\|_{L_{t,x}^{q_0}}(\|\mathcal{D}u_3\|_{L_{t,x}^{10-}}+\|\mathcal{D}u_4\|_{L_{t,x}^{10-}})^{p-2},\] which is bounded by |$K^{\sigma\gamma_0+}$| since |$u_4$| has frequency |$\geq K^{\gamma_0}$|⁠. Next, we prove that |$\mathcal{R}_1'$| satisfies better estimates; in fact, to bound |$\mathcal{R}_1$| we will write \[\nabla \mathcal{R}_1=P_K\mathcal{D}(F_{p-1}(u_1)\nabla u_1),\] and since |$u_1$| is supported in frequency |$\leq K/10$|⁠, we know actually that \[\nabla \mathcal{R}_1=P_K\mathcal{D}(P_{[K/4,K]}F_{p-1}(u_1)\cdot\nabla u_1).\] Thus by (2.15) we have \[K\|\mathcal{R}_1\|_{L_{t,x}^{10/7+}}\lesssim K^{o(1)}\|\mathcal{D}\nabla u_1\|_{L_{t,x}^{10/3}}\left\|P_{[K/4,K]}\mathcal{D}F_{p-1}(u_1)\right\|_{L_{t,x}^{5/2+}}\lesssim K^{o(1)}\sum_{M}\min(1,K^{-1}M)\|\mathcal{D}P_{M}u_1\|_{L_{t,x}^{q_0}}\] using the fact that |$\|\mathcal{D}u\|_{L_{t,x}^{10-}}\lesssim 1$|⁠. This implies |$K\|\mathcal{R}_1\|_{L_{t,x}^{10/7+}}\lesssim K^{\sigma+}$|⁠, since by Strichartz we have |$\|\mathcal{D}P_{M}u_1\|_{L_{t,x}^{q_0}}\lesssim M^{\sigma+}$|⁠. Step 3: estimate of |$\mathcal{R}_3'$|⁠. In |$\mathcal{R}_3$| we may replace |$\mathbb{P}_{\neq 0}\Omega$| by |$\mathbb{P}_{\neq 0}P_{\leq K'}\Omega$| (and |$u_2$| by |$P_{[K/4,4K]}u$|⁠), since \[\|\mathcal{D}P_{>K'}\Omega\|_{L_{t,x}^{5/2+}}\lesssim\sum_{M\geq K'}M^{o(1)}\sum_{L\leq K'}M^{-1}L\|\mathcal{D}P_Lu_3\|_{L_{t,x}^{q_0}}\lesssim \sum_{M\geq K'}M^{\sigma+}\lesssim K^{\sigma\gamma_0+}\] using the fact that |$\|\mathcal{D}u_3\|_{L_{t,x}^{10-}}\lesssim 1$|⁠, which implies \[K\left\|P_K\mathcal{D}\left(u_2\cdot P_{>K'}\Omega\right)\right\|_{L_{t,x}^{10/7+}}\lesssim \|K\mathcal{D}u_2\|_{L_{t,x}^{10/3}}\|\mathcal{D}P_{>K'}\Omega\|_{L_{t,x}^{5/2+}}\lesssim K^{\sigma\gamma_0+}.\] Let |$H(t)=\int_0^{t}h(t')\mathrm{d}t'$|⁠, |$\Omega'=\mathbb{P}_{\neq 0}P_{\leq K'}\Omega$| and |$w=K\max(1,K/N)P_{[K/4,4K]}u$|⁠, we have |$\|w\|_{X^{0,b}}\lesssim 1$| because |$\|\mathcal{D}u\|_{X^{1,b}}\lesssim1$|⁠, and |$\Omega'$| and |$w$| are compactly supported in time. To bound |$\mathcal{R}_3'$| we only need to bound \[\min(1,N/K)\|\chi(t){\rm e}^{iH(t)}P_K\mathcal{D}(w\cdot \Omega')\|_{X^{0,b-1}}\sim \|\chi(t){\rm e}^{iH(t)}P_K(w\cdot \Omega')\|_{X^{0,b-1}},\] by definition of |$\mathcal{D}$|⁠. Choose some |$z$| such that |$\|z\|_{X^{0,1-b}}\leq1 $|⁠, by duality we only need to bound the quantity \begin{equation} \mathcal{J}:=\int \chi(t){\rm e}^{iH(t)}\overline{z}\cdot P_K( w\cdot \Omega')=\sum_{k_1=k_2+k_3}\int_{\xi_1=\xi_0+\xi_2+\xi_3}\widehat{\chi e^{iH}}(\xi_0)\overline{\widehat{P_Kz}(k_1,\xi_1)}\widehat{w}(k_2,\xi_2)\widehat{\Omega'}(k_3,\xi_3). \end{equation} (5.5) We always have |$|k_3|\lesssim K^{\gamma_0}$| in the integral (5.5). Let |$P=\chi {\rm e}^{iH}$|⁠, then clearly |$P\in L^2$|⁠; moreover |$\partial_tP=(ih\chi+\chi'){\rm e}^{itH}$| also belongs to |$L^2$|⁠, since |$|h(t)|\lesssim\|u(t)\|_{L_x^{p-1}}^{p-1}\lesssim 1$| by Sobolev. This gives by Hölder that \begin{equation} \|\langle \xi\rangle^{(1/2)-}\widehat{P}(\xi)\|_{L^1}\lesssim 1. \end{equation} (5.6) Now notice that |$\gamma_1>4\gamma_0+$|⁠. If the integral (5.5) is restricted to the region |$|\xi_0|\gtrsim K^{\gamma_1}$| by inserting a suitable cut-off function |$(1-\chi)(K^{-\gamma_1}\xi_0)$|⁠, then using Hölder, the corresponding contribution will be bounded by \[|\mathcal{J}_1|\lesssim\left\|\mathcal{F}^{-1}\left(\widehat{\chi {\rm e}^{iH}}(\xi_0)(1-\chi)(K^{-\gamma_1}\xi_0)\right)\right\|_{L_{t}^{\infty}}\cdot \|P_Kz\|_{L_{t,x}^{10/3-}}\|w\|_{L_{t,x}^{10/3}}\|\Omega'\|_{L_{t,x}^{5/2+}}\lesssim K^{-\gamma_1/2+},\] since \[\left\|\widehat{\chi {\rm e}^{iH}}(\xi_0)(1-\chi)(K^{-\gamma_1}\xi_0)\right\|_{L^1}\lesssim K^{-\gamma_1/2+}\] by (5.6), and |$\|\Omega'\|_{L_{t,x}^{5/2+}}\lesssim\|u_3\|_{L_{t,x}^{5(p-1)/2+}}^{p-1}\lesssim 1$|⁠. Now we only need to study \begin{equation} \mathcal{J}_2:=\sum_{k_1=k_2+k_3}\int_{\xi_1=\xi_0+\xi_2+\xi_3}\widehat{P^*}(\xi_0)\overline{\widehat{P_Kz}(k_1,\xi_1)}\widehat{w}(k_2,\xi_2)\widehat{\Omega'}(k_3,\xi_3), \end{equation} (5.7) where \[\widehat{P^*}(\xi_0)=\widehat{\chi {\rm e}^{iH}}(\xi_0)\chi(K^{-\gamma_1}\xi_0),\] and we easily see that \[|P^*(t)|\lesssim(1+|t|)^{-10}.\] Next, if the integral (5.7) is restricted to the region |$|\xi_1+ 2\pi Q(k_1)|\gtrsim K^{\gamma_1}$| by inserting a cutoff function |$(1-\chi)(K^{-\gamma_1}(\xi_1+ 2\pi Q(k_1)))$|⁠, then we define |$z'$| by \[\widehat{z'}(k_1,\xi_1)=\widehat{P_Kz}(k_1,\xi_1)\cdot (1-\chi)(K^{-\gamma_1}(\xi_1+ 2\pi Q(k_1))),\] and use \[\|z'\|_{L_{t,x}^2}\lesssim K^{-\gamma_1(1-b)}\|z\|_{X^{0,1-b}}\lesssim K^{-\gamma_1/2+}\] to bound the corresponding contribution by \[|\mathcal{J}_3|\lesssim\|P^*\|_{L_{t}^{\infty}}\cdot \|z'\|_{L_{t,x}^{2}}\|w\|_{L_{t,x}^{10/3}}\|\Omega'\|_{L_{t,x}^{5}}\lesssim K^{-\gamma_1/2+}\|P_{\leq K'}u\|_{L_{t,x}^{5(p-1)}}^{p-1}\lesssim K^{-\gamma_1/2+(p-3)\gamma_0/2+}.\] The same estimate holds, with |$o(1)$| differences in the power of |$K$|⁠, if (5.7) is restricted to the region |$|\xi_2+ 2\pi Q(k_2)|\gtrsim K^{\gamma_1}$|⁠. Now we may replace |$P_Kz$| by |$z'':=P_Kz-z'$|⁠, and |$w$| by |$w''$| which is defined similarly, and reduce to estimating \begin{equation} \mathcal{J}_4:=\sum_{k_1=k_2+k_3}\int_{\xi_1=\xi_0+\xi_2+\xi_3}\widehat{P^*}(\xi_0)\overline{\widehat{z''}(k_1,\xi_1)}\widehat{w''}(k_2,\xi_2)\widehat{\Omega'}(k_3,\xi_3). \end{equation} (5.8) Note that |$z''$| and |$w''$| still satisfy the |$X^{0,1-b}$| and |$X^{0,b}$| bounds. Next, define the operators |$\mathcal{P}_L$| and |$\mathcal{Q}_L$| as follows: \[\widehat{\mathcal{P}_LF}(k,\xi)=\chi(L^{-1}\xi)\widehat{F}(k,\xi),\qquad\widehat{\mathcal{Q}_LF}(k,\xi)=(1-\chi(L^{-1}\xi))\widehat{F}(k,\xi),\] we decompose |$\Omega'=\Omega_1'+\Omega_2'$| where \[\Omega_1'=\frac{p+1}{2}\mathbb{P}_{\neq 0}P_{\leq K'}|\mathcal{P}_{K^{\gamma_1/2}}u_3|^{p-1},\] and \begin{align*} \Omega_2'&=\mathbb{P}_{\neq 0}P_{\leq K'}\bigg(\mathcal{Q}_{K^{\gamma_1/2}} u_3\cdot \int_0^1 F_{p-2}(\mathcal{P}_{K^{\gamma_1/2}}u_3+\zeta \mathcal{Q}_{K^{\gamma_1/2}}u_3)\,\mathrm{d}\zeta\\ &\qquad\qquad\qquad+\overline{\mathcal{Q}_{K^{\gamma_1/2}}u_3}\cdot \int_0^1 F_{p-2}(\mathcal{P}_{K^{\gamma_1/2}}u_3+\zeta \mathcal{Q}_{K^{\gamma_1/2}}u_3)\,\mathrm{d}\zeta\bigg). \end{align*} For |$\Omega_2'$| one has \begin{align} \|\Omega_2'\|_{L_{t,x}^{5/2}([0,1]\times\mathbb{T}^3)}&\lesssim\|\mathcal{Q}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{5/2}([0,1]\times\mathbb{T}^3)}(\|\mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{\infty}([0,1]\times\mathbb{T}^3)}+\|\mathcal{Q}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{\infty}([0,1]\times\mathbb{T}^3)})^{p-2}\notag\\ &\lesssim K^{-\gamma_1/5+(p-2)\gamma_0/2+}, \end{align} (5.9) since by interpolation (Namely, by interpolating between |$X^{1,0}\hookrightarrow L_{t}^2L_x^{5/2}([0,1]\times\mathbb{T}^3)$| and |$X^{1,1/2+}\hookrightarrow L_t^{\infty}L_x^{5/2}([0,1]\times\mathbb{T}^3)$|⁠.) \[\|\mathcal{Q}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{5/2}([0,1]\times\mathbb{T}^3)}\lesssim K^{o(1)}\|\mathcal{Q}_{K^{\gamma_1/2}}u_3\|_{X^{0,1/10+}}\lesssim K^{-\gamma_1/5+};\] (notice that |$\mathcal{F}\mathcal{Q}_{K^{\gamma_1/2}}u_3(k,\xi)$| is supported where |$|k|\lesssim K^{\gamma_0}$| and |$|\xi|\gtrsim K^{\gamma_1/2}$|⁠, on which |$|\xi+ 2\pi Q(k)|\gtrsim K^{\gamma_1/2}$| since |$\gamma_1>4\gamma_0+$|⁠), and by Hölder \[\|\mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{\infty}([0,1]\times\mathbb{T}^3)}+\|\mathcal{Q}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{\infty}([0,1]\times\mathbb{T}^3)}\lesssim \|u_3\|_{L_{t,x}^{\infty}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{\gamma_0/2+}.\] We also have \[\|\Omega_2'\|_{L_{t,x}^{5/2}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{-\gamma_1/5+(p-2)\gamma_0/2+}\] uniformly in |$n$|⁠, thus the contribution given by |$\Omega_2'$| is bounded by \begin{align*}|\mathcal{J}_5|& \lesssim\bigg|\int_{\mathbb{R}\times\mathbb{T}^3} P^*(t)\cdot \overline{z''}\cdot w''\cdot \Omega_2'\bigg|\\ & \lesssim\left\|\frac{z''}{1+|t|^2}\right\|_{L_{t,x}^{10/3-}(\mathbb{R}\times\mathbb{T}^3)} \left\|\frac{w''}{1+|t|^2}\right\|_{L_{t,x}^{10/3}(\mathbb{R}\times\mathbb{T}^3)}\left\| \Omega_2' \right\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)} \\ & \lesssim K^{-\gamma_1/5+(p-2)\gamma_0/2+}. \end{align*} Moreover, the term |$\Omega_1'$| can be decomposed as |$\Omega^*:=\mathcal{Q}_{K^{\gamma_1}}\Omega_1'$| and |$\Omega'':=\mathcal{P}_{K^{\gamma_1}}\Omega_1'$|⁠. For the term |$\Omega^*$| one has \[ \|\Omega^*\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{-\gamma_1}\|\partial_t \Omega_1'\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{-\gamma_1}\|\mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{10-}(\mathbb{R}\times\mathbb{T}^3)}^{p-2}\|\partial_t \mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{q_0}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{-\gamma_1/2} \] using the fact that \[\|\mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{10-}(\mathbb{R}\times\mathbb{T}^3)}\lesssim \|u_3\|_{L_{t,x}^{10-}}\lesssim 1;\quad \|\partial_t\mathcal{P}_{K^{\gamma_1/2}}u_3\|_{L_{t,x}^{q_0}(\mathbb{R}\times\mathbb{T}^3)}\lesssim K^{\gamma_1/2}\|u_3\|_{L_{t,x}^{q_0}}\lesssim K^{\gamma_1/2},\] so the corresponding contribution is \begin{align*} |\mathcal{J}_6|&\lesssim\bigg|\int_{\mathbb{R}\times\mathbb{T}^3} P^*(t)\cdot \overline{z''}\cdot w''\cdot \Omega^*\bigg|\lesssim\left\|\frac{z''}{1+|t|^2}\right\|_{L_{t,x}^{10/3-}(\mathbb{R}\times\mathbb{T}^3)} \left\|\frac{w''}{1+|t|^2}\right\|_{L_{t,x}^{10/3}(\mathbb{R}\times\mathbb{T}^3)}\left\|\Omega^*\right\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)}\\ &\lesssim K^{-\gamma_1/2+}. \end{align*} Finally, we are left with the term \begin{equation} \mathcal{J}_7:=\int_{\mathbb{R}\times\mathbb{T}^3}P^*(t)\cdot \overline{z''}\cdot w''\cdot \Omega''=\sum_{k_1=k_2+k_3}\int_{\xi_1=\xi_0+\xi_2+\xi_3}\widehat{P^*}(\xi_0)\overline{\widehat{z''}(k_1,\xi_1)}\widehat{w''}(k_2,\xi_2)\widehat{\Omega''}(k_3,\xi_3). \end{equation} (5.10) For this term, first we have \[|\mathcal{J}_7|\lesssim\left\|\frac{z''}{1+|t|^2}\right\|_{L_{t,x}^{10/3-}(\mathbb{R}\times\mathbb{T}^3)} \left\|\frac{w''}{1+|t|^2}\right\|_{L_{t,x}^{10/3}(\mathbb{R}\times\mathbb{T}^3)}\left\|\Omega''\right\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)} \lesssim K^{o(1)},\] since by Strichartz, \[\|\Omega''\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)}\lesssim \|\Omega_1'\|_{L_{t,x}^{5/2+}(\mathbb{R}\times\mathbb{T}^3)}\lesssim\left\|\mathcal{P}_{K^{\gamma_1/2}}u_3\right\|_{L_{t,x}^{5(p-1)/2+}(\mathbb{R}\times\mathbb{T}^3)}^{p-1}\lesssim\|u_3\|_{L_{t,x}^{5(p-1)/2+}}^{p-1}\lesssim 1.\] Moreover, by the definition of these factors, we know that in the |$\xi$|-integral (5.10), we must have \begin{equation} \max(|\xi_0|,|\xi_1+ 2\pi Q(k_1)|,|\xi_2+ 2\pi Q(k_2)|,|\xi_3+ 2\pi Q(k_3)|)\ll K^{\gamma_1}. \end{equation} (5.11) This gives that \[\left|Q(k_1)-Q(k_2)-Q(k_3)\right|\ll K^{\gamma_1},\] which gives that \begin{equation} |Q(k_1,k_3)|\ll K^{\gamma_1}, \end{equation} (5.12) since |$k_3=k_1-k_2$| and |$|Q(k_3)|\lesssim K^{2\gamma_0}\ll K^{\gamma_1}$|⁠, where \[Q(\ell,m):=\sum_{i=1}^3\beta_i\ell_im_i\] denotes the bilinear form corresponding to |$Q(k)$|⁠. Moreover, since |$k_3\neq 0$|⁠, we get |$k_1\in\mathcal{Y}$|⁠, where \[\mathcal{Y}:=\bigcup_{0\neq\ell\in\mathbb{Z}^3,|\ell|\lesssim K^{\gamma_0}}\big\{k\in\mathbb{R}^3:|k|\lesssim K, |Q(k,\ell)|\ll K^{\gamma_1}\big\}\subset\mathbb{R}^3\] is the union of at most |$O(K^{3\gamma_0})$| rectangular cuboids of dimensions |$K\times K\times O(K^{\gamma_1})$|⁠. This completes the estimate for |$\mathcal{R}_3'$|⁠. The estimate for |$\mathcal{R}_2'$| is done in completely analogous way; in fact, one may first replace the |$F_{p-1}(u_1)$| factor by |$P_{\leq K/10}F_{p-1}(u_1)$|⁠, then argue in exactly the same way as above, the only difference being that we now have |$\xi_1+\xi_2=\xi_0+\xi_3$| in the integral (5.5) due to the presence of |$\overline{u_2}$|⁠, \[\max(|\xi_0|,|\xi_1+ 2\pi Q(k_1)|,|\xi_2+2\pi Q(k_2)|,|\xi_3+2\pi Q(k_3)|)\gtrsim K^{2}\] is always true. Step 4: from the estimates on |$\mathcal{R}$| to the desired inequality. Summing up, we get that |$\mathcal{R}'$| can be decomposed into two parts, \[\mathcal{R}'=\mathcal{R}''+\widetilde{\mathcal{R}},\] where \[\left\|\chi(t)\mathcal{R}''\right\|_{X^{1,b-1}}\lesssim K^{o(1)}K^{\max(\sigma\gamma_0,-\gamma_1/5+(p-2)\gamma_0/2)},\] and \[\|\chi(t)\widetilde{\mathcal{R}}\|_{X^{1,b-1}}\lesssim K^{o(1)} ,\quad\mathrm{supp}(\mathcal{F}\widetilde{\mathcal{R}})\subset\mathcal{Y}\times\mathbb{R}.\] Using Proposition 2.1, one gets that \begin{equation} \left\|v(t)-{\rm e}^{it\Delta_{\beta}}v(0)\right\|_{L_{t,x}^{q_0}([0,\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+}\cdot\max\left(K^{\sigma\gamma_0},K^{-\gamma_1/5+(p-2)\gamma_0/2},K^{3\gamma_0-(p-3)(1-\gamma_1)/6}\right). \end{equation} (5.13) Optimizing the right hand side leads to the choice |$\gamma_0 = \frac{2(p-3)}{3(3p+7)}$| and |$\gamma_1 = 7.5 \gamma_0$|⁠, which gives \begin{equation} \left\|v(t)-{\rm e}^{it\Delta_{\beta}}v(0)\right\|_{L_{t,x}^{q_0}([0,\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+\theta_1+}. \end{equation} (5.14) By time translation, one also gets that \begin{equation} \left\|v(t)-{\rm e}^{i(t-m\varepsilon)\Delta_{\beta}}v(m\varepsilon)\right\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+\theta_1+}. \end{equation} (5.15) for each |$0\leq m\leq K^{\gamma}$|⁠. Moreover, using the same arguments as above, one can also prove that \begin{equation} \sup_{n\in\mathbb{Z}}\left\|{\rm e}^{i(n\varepsilon+t)\Delta_{\beta}}\left[v((m+1)\varepsilon)-{\rm e}^{i\varepsilon\Delta_{\beta}}v(m\varepsilon)\right]\right\|_{L_{t,x}^{q_0}([0,\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+\theta_1+} \end{equation} (5.16) for each |$0\leq m\leq K^{\gamma}$|⁠. In fact, by time translation we may assume |$m=0$|⁠, so \[v(\varepsilon)-e^{i\varepsilon\Delta_\beta}v(0)=-i\int_0^\varepsilon {\rm e}^{i(\varepsilon-t')\Delta_{\beta}}\mathcal{R}'(t')\,\mathrm{d}t'.\] Using Proposition 2.1, part (3), and the decomposition |$\mathcal{R}'=\widetilde{R}+\mathcal{R}''$| above, we can decompose |$v(\varepsilon)-{\rm e}^{i\varepsilon\Delta_{\beta}}v(0)$| into two terms, one having |$H^1$| norm bounded by \[K^{o(1)}K^{\max(\sigma\gamma_0,-\gamma_1/5+(p-2)\gamma_0/2)},\] the other having bounded |$H^1$| norm and Fourier transform supported in |$\mathcal{Y}$|⁠. Then, (5.16) follows from Strichartz. Combining (5.15) and (5.16), one easily gets that \begin{align} \sum_{m=0}^{K^{\gamma}}\left\|v\right\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}&\lesssim K^{\sigma+\theta_1+\gamma+}+\sum_{m=0}^{K^\gamma}\|{\rm e}^{i(t-m\varepsilon)\Delta_\beta}v(m\varepsilon)\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\notag\\ &\lesssim K^{\sigma+\theta_1+\gamma+}+\sum_{m=0}^{K^{\gamma}}\|{\rm e}^{it\Delta_{\beta}}P_K\mathcal{D}u(0)\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\notag\\ &\quad+\sum_{m=0}^{K^{\gamma}}\sum_{j=0}^{m-1}\|{\rm e}^{i(t-(j+1)\varepsilon)\Delta_\beta}v((j+1)\varepsilon)-{\rm e}^{i(t-j\varepsilon)\Delta_\beta}v(j\varepsilon)\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\notag\\ &\lesssim K^{\sigma+\theta_1+2\gamma+}+\sum_{m=0}^{K^{\gamma}}\|{\rm e}^{it\Delta_{\beta}}P_K\mathcal{D}u(0)\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}. \end{align} (5.17) By the long-time Strichartz estimate (part (6) of Proposition 2.1) combined with Hölder in |$m$|⁠, one gets that \[\sum_{m=0}^{K^{\gamma}}\left\|v\right\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\lesssim K^{\sigma+\theta_1+2\gamma+}+K^{\sigma+\gamma-\gamma/q_0+}.\] This completes the proof. ■ Proof of Theorem 1.3 in the irrational case. By Proposition 5.1, choosing |$\gamma=q_0|\theta_1|/(q_0+1)$|⁠, one has that, as long as |$T>K^{|\theta_1|+}$| and |$E[\mathcal{D}u(t)]\lesssim 1$| for all |$t\in[0,T]$|⁠, \[\sum_{m=0}^{\varepsilon^{-1}T}\|P_K\mathcal{D}u\|_{L_{t,x}^{q_0}([m\varepsilon,(m+1)\varepsilon]\times\mathbb{T}^3)}\lesssim TK^{\sigma+}\cdot K^{\frac{\theta_1}{q_0+1}} .\] Given initial data, choose |$N$| such that |$N\sim A$| and |$E[\mathcal{D}u(0)]\leq 10E$|⁠. By Propositions 4.1, as long as \[\sup_{0\leq t\leq T}E[\mathcal{D}u(t)]\leq 20E\] for all |$t\in[0,T]$|⁠, one has that \begin{align*}\sup_{0\leq t\leq T}E[\mathcal{D}u(t)] - 10E & \lesssim N^{\max(p-5,-1)}T+N^{o(1)}\sum_K K^{o(1)}\min(1,N^{-1}K)\cdot TK^{\sigma+}\cdot K^{\frac{\theta_1}{q_0+1}}\\ & \lesssim N^{\max(p-5,-1)}T+N^{\sigma+\frac{\theta_1}{q_0+1}+}T.\end{align*} By a bootstrap argument, this gives that |$E[\mathcal{D}u(t)]\lesssim 1$| and |$\|u(t)\|_{H^2}\lesssim A$|⁠, up to time \[T=N^{\frac{5-p}{2}+\frac{\theta(p)}{2}},\] since we have \[\theta(p)=\frac{\max(p-3,5-p)}{182}\leq \frac{363}{364}\cdot\frac{2}{3}\frac{(6-p)(p-3)(5-p)}{(16-p)(3p+7)}<\frac{2|\theta_1|}{q_0+1}\] by elementary computations. By the same argument as in the general case, this implies (1.3). ■ Funding This work was partially supported by National Science Foundation [DMS-1501019 to P. G.]. 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König,, Joachim;Legrand,, François;Neftin,, Danny

doi: 10.1093/imrn/rny016pmid: N/A

Abstract Given a field k of characteristic zero and an indeterminate T over k, we investigate the local behavior at primes of k of finite Galois extensions of k arising as specializations of finite Galois extensions E/k(T) (with E/k regular) at points |$t_{0} \in \mathbb{P}^{1}(k)$|⁠. We provide a general result about decomposition groups at primes of k in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert-Grunwald property, and finite parametric sets. 1 Introduction 1.1 Grunwald problems Given a number field k, Grunwald problems concern the structure of completions |$L_{\mathfrak{p}}/k_{\mathfrak{p}}$| of G-extensions L/k at any finitely many given primes |${\mathfrak{p}}$| of k. Here, a G-extension L/k is a finite Galois extension of k with Galois group G, and |$k_{\mathfrak{p}}$| (resp., |$L_{\mathfrak{p}}$|⁠) denotes the completion of k (resp., of L) at |${\mathfrak{p}}$| (resp., at a prime |${\mathfrak{P}}$| of L lying over |${\mathfrak{p}}$|⁠). Recall that the latter is independent of the choice of |${\mathfrak{P}}$| (up to |$k_{\mathfrak{p}}$|-isomorphism). The original motivation for such problems arose from their key role in the structure theory of finite dimensional division algebras over number fields. Since then, they have received much attention, especially due to connections with the regular inverse Galois problem; see [10, 11], and with weak approximation; see, for example, [9, 12, 16, 23]. More precisely, given a finite set |$\mathcal{S}$| of primes of k, and given finite Galois extensions |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$|⁠, |${\mathfrak{p}} \in \mathcal{S}$|⁠, with Galois groups embedding into G, the Grunwald problem |$(G, (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$| asks whether there is a G-extension L/k whose completion |$L_{\mathfrak{p}}$| at |${\mathfrak{p}}$| is |$k_{\mathfrak{p}}$|-isomorphic to |$L^{({\mathfrak{p}})}$|⁠, |${\mathfrak{p}} \in \mathcal{S}$|⁠. If such an extension L/k exists, it is called a solution to the Grunwald problem |$(G, (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}} )$|⁠. We note that, instead of prescribing the local extensions |$L_{{\mathfrak{p}}}/k_{\mathfrak{p}}$|⁠, |${\mathfrak{p}}\in \mathcal{S}$|⁠, weaker versions ask for the existence of a G-extension L/k with prescribed local degrees |$[L_{\mathfrak{p}}:k_{\mathfrak{p}}]$|⁠, |${\mathfrak{p}}\in \mathcal{S}$|⁠, or with prescribed local Galois groups |$\textrm{Gal}(L_{\mathfrak{p}}/k_{\mathfrak{p}})$|⁠, |${\mathfrak{p}}\in \mathcal{S}$|⁠. These weaker versions usually suffice for applications to classical problems, see, for example, [38, Corollary 2] and [33]. Examples of Grunwald problems |$(G,(L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$| with no solution L/k occur already for cyclic groups G, when |$\mathcal{S}$| contains a prime of k lying over 2 [37]. However, it is expected [16, Section 1] that, for solvable groups G, every Grunwald problem |$(G,(L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$| has a solution, provided |$\mathcal{S}$| is disjoint from some finite set |$\mathcal{S}_{\textrm{{exc}}}$| of “exceptional” primes of k, depending only on G and k. This is known when (1) G is abelian, and |$\mathcal{S}_{\textrm{{exc}}}$| is the set of primes of k dividing 2 [28, (9.2.8)]; (2) G is an iterated semidirect product |$A_{1} \rtimes \left (A_{2} \rtimes \cdots \rtimes A_{n}\right )$| of finite abelian groups, and |$\mathcal{S}_{\textrm{{exc}}}$| is the set of primes of k dividing |G|; see [16, Théorème 1] and [12, Theorem 1.1]; (3) G is solvable of order prime to the number of roots of unity in k, and |$\mathcal{S}_{\textrm{{exc}}}=\emptyset $| [28, (9.5.5)]; and (4) there exists a generic extension for G over k, and |$\mathcal{S}_{\textrm{{exc}}}=\emptyset $| [32, Theorem 5.9]. Among the above, the latter is the only method which applies to non-solvable groups. However, the family of non-solvable groups for which a generic extension is known is quite restricted, for example, it is unknown whether the alternating group An has a generic extension for n ≥ 6. See [17] for an overview on generic extensions. The main source of realizations of non-solvable groups G over k is via k-regular G-extensions, that is, via G-extensions E/k(T), where T is an indeterminate over k and k is algebraically closed in E. Indeed, by Hilbert’s irreducibility theorem, every nontrivial k-regular G-extension E/k(T) has infinitely many linearly disjoint specializations |$E_{t_{0}}/k$|⁠, |$t_{0}\in \mathbb{P}^{1}(k)$|⁠, with Galois group G. Many groups have been realized by this method; see, for example, [27], and references within, as well as [40] for more recent examples. This specialization process provides a natural way to attack Grunwald problems for k-regular Galois groups, that is, for finite groups G admitting a k-regular G-extension of k(T). Namely, given such an extension E/k(T), it is natural to ask for the local behavior of specializations |$E_{t_{0}}/k$|⁠, |$t_{0}\in \mathbb{P}^{1}(k)$|⁠. That is, which local extensions |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$|⁠, which local Galois groups |$\textrm{Gal}\left (L^{({\mathfrak{p}})}/k_{\mathfrak{p}}\right )$|⁠, and which local degrees |$\left [L^{({\mathfrak{p}})}:k_{\mathfrak{p}}\right ]$| arise by completing the specialization |$E_{t_{0}}/k$| at primes |${\mathfrak{p}}$| of k, when t0 runs over |$ \mathbb{P}^{1}(k)$|? For points |$t_{0}\in \mathbb{P}^{1}(k)$| which are |${\mathfrak{p}}$|-adically far from branch points of E/k(T), this approach was deeply investigated by Dèbes and Ghazi [10, 11] (see [11, Section 1.6] for a review of related previous results) and applies only to unramified local extensions. Namely, given a k-regular G-extension E/k(T), a finite set |$\mathcal{S}$| of primes of k, disjoint from some finite set |$\mathcal{S}_{\textrm{{exc}}}:=\mathcal{S}_{\textrm{{exc}}}(E/k(T))$| (depending only on E/k(T)), and unramified extensions |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| with Galois group embedding into G, |${\mathfrak{p}}\in \mathcal{S}$|⁠, [11] provides t0 ∈ k such that |$E_{t_{0}}/k$| is a solution to the Grunwald problem |$(G,(L^{({\mathfrak{p}})}/k_{\mathfrak{p}} )_{{\mathfrak{p}}\in \mathcal{S}} )$|⁠. 1.2 Main results The goal of this paper is to study the local behavior at a prime |${\mathfrak{p}}$| of k of specializations |$E_{t_{0}}/k$|⁠, when t0 is |${\mathfrak{p}}$|-adically close to a branch point of E/k(T). 1.2.1 Background A first related conclusion can be derived from the algebraic cover theory of Grothendieck. Namely, if |${\mathfrak{p}}$| is not in some finite set |$\mathcal{S}_{\textrm{{exc}}}:=\mathcal{S}_{\textrm{{exc}}}(E/k(T))$| of primes of k and if |${\mathfrak{p}}$| ramifies in the specialization |$E_{t_{0}}/k$|⁠, then, t0 and some branch point ti of E/k(T) meet modulo |${\mathfrak{p}}$|. Note that the set |$\mathcal{S}_{\textrm{{exc}}}$| is chosen such that every t0 meets at most one branch point modulo |${\mathfrak{p}}$|⁠. In the special case where ti is k-rational and |$v_{\mathfrak{p}}(t_{0})$| is non negative, this means that |$a_{\mathfrak{p}}:=v_{\mathfrak{p}}\left (t_{0}-t_{i}\right )$| is positive, where |$v_{\mathfrak{p}}$| denotes the normalized |${\mathfrak{p}}$|-adic valuation. A fundamental theorem by Beckmann [4, 7] then asserts that the inertia group |$I_{t_{0},{\mathfrak{p}}}$| of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| is determined by |$a_{\mathfrak{p}}$| and the inertia group |$I_{t_{i}}$| at a fixed prime of E lying over the prime |$\mathcal{P}_{i}$| of |$k\left [T-t_{i}\right ]$| generated by T − ti. Namely, |$I_{t_{0},{\mathfrak{p}}}$| is conjugate to |$I_{t_{i}}^{a_{\mathfrak{p}}}$|⁠. We refer to Section 3 for more details. 1.2.2 Decomposition groups of specializations Given |$t_{0}\in \mathbb{P}^{1}(k)$|⁠, assumed to meet the branch point ti modulo |${\mathfrak{p}}$|⁠, we show that the entire decomposition group at |${\mathfrak{p}}$| of the specialization |$E_{t_{0}}/k$| is determined by the local behavior at ti, thus extending Beckmann’s theorem. Namely, suppose for simplicity that ti is k-rational, and let |$D_{t_{i}}$| denote the geometric decomposition group at a fixed prime of E lying over the geometric prime |$\mathcal{P}_{i}$|⁠. Recall that the specialization |$E_{t_{i}}/k$| has Galois group |$D_{t_{i}}/I_{t_{i}}$|⁠; let |$\varphi _{i}:D_{t_{i}}\rightarrow D_{t_{i}}/I_{t_{i}}$| be the natural projection. Let |$D_{t_{i},{\mathfrak{p}}}$| be the decomposition group at a prime |${\mathfrak{P}}$| of |$E_{t_{i}}$| lying over |${\mathfrak{p}}$|⁠; this is a subgroup of |$D_{t_{i}}/I_{t_{i}}$|⁠. Note that, up to conjugation, the subgroup |$\varphi _{i}^{-1}(D_{t_{i},{\mathfrak{p}}})$| of G is independent of the choice of an arithmetic prime |${\mathfrak{P}}$| lying over |${\mathfrak{p}}$| and a geometric prime lying over |$\mathcal{P}_{i}$|⁠. Theorem 1.1. There exists a finite set |$\mathcal{S}_{\textrm{{exc}}}:=\mathcal{S}_{\textrm{{exc}}}(E/k(T))$| of primes of k that satisfies the following property. Suppose that the given prime |${\mathfrak{p}}$| of k is outside |$\mathcal{S}_{\textrm{{exc}}}$|⁠. Moreover, suppose that the given branch point ti is k-rational, that t0 and ti meet modulo |${\mathfrak{p}}$|⁠, and that the exponent |$a_{\mathfrak{p}}$| is coprime to |$|I_{t_{i}}|$|⁠. Then, the decomposition group of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| is conjugate by an element of G to |$\varphi _{i}^{-1}(D_{t_{i},{\mathfrak{p}}} )$|⁠. Note that |$\varphi _{i}^{-1}(D_{t_{i},{\mathfrak{p}}})$| is the group generated by |$I_{t_{i}}$| and a lift of the Frobenius of |$E_{t_{i}}/k$| at |${\mathfrak{p}}$|⁠. We refer to Theorem 4.1 for a more general version of Theorem 1.1, where we relax the assumptions on ti and |$a_{\mathfrak{p}}$|⁠, and which is stated over more general base fields. 1.2.3 Solving Grunwald problems Theorem 1.1 shows that the decomposition group of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| is determined by the local data |$(\varphi _{i},D_{t_{i},{\mathfrak{p}}})$| at ti, when t0 and ti meet modulo |${\mathfrak{p}}$|⁠. Theorem 1.2 below shows that this is the only constraint on completions of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| for such specialization points t0. Theorem 1.2. Suppose that |$\mathcal{S}$| is a finite set of primes of k disjoint from some finite set of primes |$\mathcal{S}^\prime _{\textrm{{exc}}}:=\mathcal{S}^\prime _{\textrm{{exc}}}(E/k(T))$|⁠. For each |${\mathfrak{p}}\in \mathcal{S}$|⁠, fix a k-rational branch point |$t_{i({\mathfrak{p}})}$| of E/k(T) and a finite Galois extension |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| with Galois group (resp., inertia group) |$\varphi _{i({\mathfrak{p}})}^{-1}(D_{t_{i({\mathfrak{p}})},{\mathfrak{p}}})$| (resp., |$I_{t_{i({\mathfrak{p}})}}$|⁠). Then, there exists t0 ∈ k such that |$E_{t_{0}}/k$| is a solution to the Grunwald problem |$(G,(L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$|⁠. See Theorem 4.4 for a more general version of Theorem 1.2, where the assumption on the branch points is relaxed, and where k is not necessarily a number field. Given a single homomorphism |$\varphi _{i}:D_{t_{i}}\rightarrow D_{t_{i}}/I_{t_{i}}$| at a (k-rational) branch point ti of E/k(T), Theorem 1.2 provides a solution to all Grunwald problems |$(G, (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$|⁠, where, for each |${\mathfrak{p}}$| in |$\mathcal{S}$|⁠, the local extension |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| has inertia group |$I_{t_{i}}$| and decomposition group |$\varphi _{i}^{-1}(D_{t_{i},{\mathfrak{p}}})$|⁠. Such results are especially applicable to problems where the primes are allowed to vary and the decomposition groups are fixed; see, in particular, Section 1.3.1. 1.2.4 About the proof Let R be the ring of integers of k. Our approach to Theorems 1.1 and 1.2 considers the fraction field F of the two-dimensional domain |$R_{\mathfrak{p}}\left [\left [T-t_{i}\right ]\right ]$|⁠, which is also the completion of R[T] at the ideal generated by |${\mathfrak{p}}$| and T − ti. We use a theorem of Eisenstein, recalled as Theorem 2.4, to show that, for all but finitely many primes |${\mathfrak{p}}$| of k, the group Gal(E ⋅ F/F) is determined by the local data |$(\varphi _{i},D_{t_{i},{\mathfrak{p}}})$| at ti. On the other hand, we show that, by reducing the extension E ⋅ F/F at a place extending the place (T − t0) of k(T), the resulting extension of residue fields is the completion of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$|⁠, giving the desired connection between |$D_{t_{0},{\mathfrak{p}}}$| and the local data at ti. See Section 4.2. The above method is considerably different from that of Dèbes and Ghazi, which, to our knowledge, does not apply to ramified extensions; see [11, Section 1.6], and which is based on specializations of extensions of |$k_{\mathfrak{p}}(T)$|⁠. We use the theory of Brauer embedding problems to determine the extensions |$M/k_{\mathfrak{p}}$| which satisfy the constraints imposed by Theorem 1.1 on completions of specializations |$(E_{t_{0}})_{\mathfrak{p}}$|⁠, and the above reduction process to show that such fields M are obtained as specializations |$(E_{t_{0}})_{\mathfrak{p}}$|⁠. 1.3 Applications Theorem 1.1 is then used to study crossed product division algebras, the Hilbert-Grunwald property, and finite parametric sets. 1.3.1 Prescribing decomposition groups and crossed products Recall that the Galois group D of a tamely ramified extension of |$k_{\mathfrak{p}}$| is metacyclic, that is, D admits a cyclic normal subgroup I such that D/I is cyclic. The following application of Theorem 1.1 ensures the appearance of certain metacyclic subgroups of G as decomposition groups of specializations of the k-regular G-extension E/k(T). Theorem 1.3. Given a branch point ti of E/k(T), let |$D_{t_{i}}$| (resp., |$I_{t_{i}}$|⁠) denote the decomposition (resp., inertia) group at ti, and let τ be an element of |$D_{t_{i}}$|⁠. Then, there exist infinitely many primes |${\mathfrak{p}}$| of k, and, for each such prime |${\mathfrak{p}}$|⁠, infinitely many t0 ∈ k such that |$E_{t_{0}}/k$| is a G-extension with decomposition group |$\left \langle I_{t_{i}},\tau \right \rangle $| at |${\mathfrak{p}}$|⁠. We refer to Theorem 4.6 for a generalization of Theorem 1.3, where we consider finitely many primes of k at the same time. We then apply this result to prove the existence of G-crossed product division algebras over number fields for various finite groups G. Recall that a finite dimensional division algebra over its center k is a G-crossed product if it admits a maximal subfield L that is finite and Galois over k, and such that Gal(L/k) = G. A G-crossed product division algebra is equipped with an explicit structure, which plays a key role in the theory of central simple algebras. By Schacher’s result [33, Theorem 4.1], the existence of a G-crossed product division algebra with center |$\mathbb Q$| implies that G has metacyclic Sylow subgroups. The converse is a long standing open conjecture, originating in [33]: for every finite group G with metacyclic Sylow subgroups, there exists a G-crossed product division algebra with center |$\mathbb Q$|⁠. Although this conjecture has been extensively studied, cf. [1, Section 11.A], the only finite non-abelian simple groups for which the conjecture is known to hold are |$A_{5} \cong{\textrm{{PSL}}}_{2}\left (\mathbb{F}_{4}\right ) \cong{\textrm{{PSL}}}_{2}\left (\mathbb{F}_{5}\right )$| [15, Theorem 1], |$A_{6} \cong{\textrm{{PSL}}}_{2}\left (\mathbb{F}_{9}\right )$| [14, Theorem 6], A7 [14, Theorem 6], |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{7}\right )$| [2, Proposition 5], |${\textrm{{PSL}}}_{2}(\mathbb{F}_{11})$| [2, Theorem 2], and the Mathieu group M11 [20]. Given a k-regular G-extension E/k(T) with suitable properties, we use Theorem 1.3 to establish a general criterion for the existence of specialization points t0 ∈ k such that |$E_{t_{0}}$| is a maximal subfield of a G-crossed product. We refer to Theorem 5.2 for our precise result. We then use this criterion to derive the first infinite family of finite non-abelian simple groups G with a G-crossed product division algebra over |$\mathbb{Q}$|⁠. Theorem 1.4. Let p be a prime number such that either p ≡ 3 (mod 8) or p ≡ 5 (mod 8). Then, there exists a |$\textrm{PSL}_{2}\left({\mathbb{F}}_{p}\right)$|-crossed product division algebra with center |$\mathbb{Q}$|⁠. See Theorem 5.3 where we show more generally that Theorem 1.4 holds over an arbitrary number field k (instead of |$\mathbb{Q}$|⁠), provided primitive 4-th roots of unity are not in k. See also Theorem 5.4 where a similar conclusion is shown to hold for prime numbers lying in other arithmetic progressions. 1.3.2 On the Hilbert-Grunwald property Following a terminology due to Dèbes and Ghazi and motivated by the partial positive answer provided by Theorem 1.2, one may ask whether a given k-regular G-extension E/k(T) has the Hilbert-Grunwald property, that is, whether there exists a finite set |$\mathcal{S}_{\textrm{{exc}}}$| of primes of k such that every Grunwald problem |$(G, (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$|⁠, with |$\mathcal{S}$| disjoint from |$\mathcal{S}_{\textrm{{exc}}}$|⁠, has a solution inside the set of specializations of E/k(T). Recall that this property is always fulfilled, provided we restrict to unramified Grunwald problems [10, 11]. In contrast, our results on the local behavior of (ramified) specializations show in particular that, for many finite groups G, no k-regular G-extension of k(T) has the Hilbert-Grunwald property: Theorem 1.5. Assume that G has a non cyclic abelian subgroup. Then, no k-regular G-extension of k(T) has the Hilbert-Grunwald property. We refer to Theorem 6.2 for a more general version with finitely many k-regular G-extensions at the same time. Groups with a non cyclic abelian subgroup have been classified in the literature relatively explicitly; see, for example, the classical papers [39] and [36]. Examples of such groups contain, in addition to the obvious Sn for n ≥ 4 and Dn with n even, all non-abelian simple groups; see the proof of Corollary 6.5. In particular, these groups satisfy the conclusion of Theorem 1.5. Recall that there are many non-abelian simple groups G such that the only known realizations of G over k are specializations of k-regular G-extensions of k(T), and only finitely many k-regular G-extensions of k(T) are known (of course, up to obvious manipulations such as changes of variable, or translates by rational extensions k(s)/k(t), both of which do not yield new specializations). For such finite groups G, Theorem 1.5 implies that one cannot solve all Grunwald problems for the group G over the number field k by using only the currently known realizations of G over k. 1.3.3 Nonexistence of finite parametric sets As in [18], given a finite group G, we call a set S of k-regular G-extensions of k(T) parametric if every G-extension of k occurs as a specialization of some extension E/k(T) in S; see Definition 7.1. If S consists of a single extension E/k(T), the extension E/k(T) is called parametric. This is a generalization of the Beckmann–Black problem (over k), which, with our phrasing, asks whether every finite group G has a parametric set over k. See, for example, the survey paper [8] for more on the Beckmann–Black problem. Here, we ask whether a given finite group G has a finite parametric set over k. On the one hand, a given finite group G has a parametric extension over k, provided G has a one parameter generic polynomial over k. However, this latter condition is very restrictive. For example, in the case |$k=\mathbb{Q}$|⁠, only the subgroups of the symmetric group S3 have a one parameter generic polynomial; see [17, Sections 2.1 and 8.2]. Note that no finite group G with a finite parametric set over k, but with no one parameter generic polynomial over k, is available in the literature. On the other hand, no finite group G was known to have no finite parametric set over the given number field k, until a recent joint work by the first two authors [18]. However, the “global” strategy developed in that paper requires G to have a nontrivial proper normal subgroup which satisfies some further properties. In particular, this cannot be used for finite simple groups. As a further application of our “local” results, we obtain the first examples of finite non-abelian simple groups without finite parametric sets: Theorem 1.6. Let n ≥ 4 be an integer. Then, the alternating group An has no finite parametric set over k. We refer to Theorem 7.2 where we show more generally that a given finite group G has no finite parametric set over the number field k, provided G has a non cyclic abelian subgroup and some very weak Grunwald property holds. This last property holds in particular if G = An, by Mestre [26]; see Corollary 7.3. 2 Notation and Preliminaries The section is organized as follows. Section 2.1 is devoted to standard background on Dedekind domains, while we recall classical material on function field extensions in Section 2.2. 2.1 Basics on Dedekind domains For more on below, we refer to [34]. 2.1.1 Residue, localization, and completion Let R be a domain of characteristic zero, and k its fraction field. Given a prime|${\mathfrak{p}}$| of k, that is, given a non zero prime ideal |${\mathfrak{p}}$| of R, the residue field|$\overline{k}_{\mathfrak{p}}$| of R at |${\mathfrak{p}}$| is the fraction field |${\textrm{{Frac}}}\left (R/{\mathfrak{p}}\right )$| of |$R/{\mathfrak{p}}$|⁠. We shall assume that |$\overline{k}_{\mathfrak{p}}$| is perfect. The local ring |$R_{({\mathfrak{p}})}:=\{x/y \, : \, (x,y) \in R^{2} \, \,{\textrm{{and}}} \, \, y \notin{\mathfrak{p}}\}$| is the localization of R at |${\mathfrak{p}}$|⁠. The unique maximal ideal of |$R_{({\mathfrak{p}})}$| is |${\mathfrak{p}} R_{({\mathfrak{p}})}$|⁠, and the corresponding residue field |$R_{({\mathfrak{p}})} / {\mathfrak{p}} R_{({\mathfrak{p}})}$| is canonically isomorphic to |$\overline{k}_{\mathfrak{p}}$|⁠. If |${\mathfrak{p}} R_{({\mathfrak{p}})}$| is principal, there is a discrete valuation |$v_{\mathfrak{p}}$| on k whose valuation ring is |$R_{({\mathfrak{p}})}$|⁠. Let |$k_{\mathfrak{p}}$| be the completion of k with respect to |$v_{\mathfrak{p}}$|⁠. There is a unique valuation on |$k_{\mathfrak{p}}$| extending |$v_{\mathfrak{p}}$|⁠, the residue field of which coincides with |${\overline{k}}_{\mathfrak{p}}$| (up to canonical isomorphism). 2.1.2 Dedekind domains From now on, we assume that R is a Dedekind domain. Let L/k be a finite extension. The integral closure S of R in L is also a Dedekind domain. Let |${\mathfrak{P}}$| be a prime of L lying over |${\mathfrak{p}}$|⁠. The residue field |$\overline{L}_{\mathfrak{P}}$| (⁠|$=S/{\mathfrak{P}}$|⁠) is a finite extension of |$\overline{k}_{\mathfrak{p}}$|⁠. The residue degree of L/k at |$\mathfrak{P}$| is the degree |$f_{{\mathfrak{P}}|{\mathfrak{p}}}:= [\overline{L}_{\mathfrak{P}}:\overline{k}_{\mathfrak{p}}]$| of this finite extension. The maximal positive integer |$e:=e_{{\mathfrak{P}}|{\mathfrak{p}}}$| for which |${\mathfrak{P}}^{e}$| is contained in |${\mathfrak{p}} S$| is the ramification index of |${\mathfrak{P}}$| in L/k. The prime |${\mathfrak{p}}$| is ramified in L/k if there exists a prime |${\mathfrak{P}}$| lying over it with ramification index |$e_{{\mathfrak{P}}|{\mathfrak{p}}}>1$|⁠, and unramified otherwise. It is totally ramified if |$e_{{\mathfrak{P}}|{\mathfrak{p}}}=[L:k]$| for the unique prime |${\mathfrak{P}}$| lying over |${\mathfrak{p}}$|⁠, and totally split in L/k if |${\mathfrak{p}}$| is unramified in L/k, and if |$f_{{\mathfrak{P}}|{\mathfrak{p}}}=1$| for every prime |${\mathfrak{P}}$| of L lying over |${\mathfrak{p}}$|⁠. From now on, we assume that the extension L/k is Galois with Galois group G. The subgroup of G which consists of all elements σ such that |$\sigma \left (\mathfrak P\right )= \mathfrak P$| is the decomposition group|$D_{\mathfrak{P}}$| of L/k at |$\mathfrak P$|⁠. The residue extension |$\overline{L}_{\mathfrak{P}}/\overline{k}_{\mathfrak{p}}$| is Galois, and the restriction map |$D_{\mathfrak{P}} \rightarrow{\textrm{{Gal}}}(\overline{L}_{\mathfrak{P}}/\overline{k}_{\mathfrak{p}})$| is an epimorphism, whose kernel is the inertia group|$I_{\mathfrak{P}}$| of L/k at |${\mathfrak{P}}$|⁠. The decomposition groups |$D_{\mathfrak P_{1}}$| and |$D_{\mathfrak P_{2}}$| (resp., the inertia groups |$I_{\mathfrak P_{1}}$| and |$I_{\mathfrak P_{2}}$|⁠) of two primes |${\mathfrak{P}}_{1}$| and |${\mathfrak{P}}_{2}$| lying over |${\mathfrak{p}}$| in L/k are conjugate in G. When a prime |${\mathfrak{P}}$| lying over |${\mathfrak{p}}$| is fixed, we set |$D_{\mathfrak{p}}:=D_{{\mathfrak{P}}}$| and |$I_{\mathfrak{p}}:=I_{{\mathfrak{P}}}$|⁠. One has |$I_{\mathfrak{p}} \trianglelefteq D_{\mathfrak{p}}$|⁠, the cardinalities |$|I_{\mathfrak{p}}|$| and |$|D_{\mathfrak{p}}|$| are independent of the choice of the prime |${\mathfrak{P}}$| lying over |${\mathfrak{p}}$|⁠, and are equal to |$e_{{\mathfrak{P}}|{\mathfrak{p}}}$| and |$e_{{\mathfrak{P}}|{\mathfrak{p}}} f_{{\mathfrak{P}}|{\mathfrak{p}}}$|⁠, respectively. Similarly, the residue extension |$\overline{L}_{{\mathfrak{P}}}/\overline{k}_{\mathfrak{p}}$| of L/k at |${\mathfrak{P}}$| does not depend on the choice of |${\mathfrak{P}}$| (up to |$\overline{k}_{\mathfrak{p}}$|-isomorphism); we therefore denote it by |$\overline{L}_{{\mathfrak{p}}}/\overline{k}_{\mathfrak{p}}$|⁠. The following basic lemma describes explicitly the residue field |$\overline{L}_{\mathfrak{p}}$|⁠. Lemma 2.1. Let f(X) ∈ R[X] be a monic separable polynomial of positive degree n and splitting field L over k. Denote the roots of f(X) by |$y_{1},\dots ,y_{n}$|⁠. Let |${\mathfrak{p}}$| be a prime of k such that the discriminant of f is not contained in |${\mathfrak{p}}$|⁠. Then, one has |$\overline{L}_{\mathfrak{p}}= \overline{k}_{\mathfrak{p}}\left (\overline{y_{1}},\dots ,\overline{y_{n}}\right ),$| where |$\overline{x}$| denotes the reduction of x ∈ L modulo a prime of L lying over |${\mathfrak{p}}$|⁠. The completion |$L_{\mathfrak{P}}$| of L at |${\mathfrak{P}}$| is Galois over |$k_{\mathfrak{p}}$|⁠, with Galois group canonically isomorphic to |$D_{\mathfrak{P}}$| (via restriction from |$L_{\mathfrak{P}}$| to L). One has |$L_{\mathfrak{P}} = L \cdot k_{\mathfrak{p}}$|⁠, where |$L \cdot k_{\mathfrak{p}}$| denotes the compositum of L and |$k_{\mathfrak{p}}$| inside a given algebraic closure |$\widetilde{k_{\mathfrak{p}}}$| of |$k_{\mathfrak{p}}$|⁠. Since, up to |$k_{\mathfrak{p}}$|-isomorphism, this does not depend on the choice of |${\mathfrak{P}}$|⁠, one can speak of the completion of L/k at |${\mathfrak{p}}$|⁠, and denote it by |$L_{\mathfrak{p}} / k_{\mathfrak{p}}$|⁠. The maximal subextension of |${L_{\mathfrak{p}}} /{k_{\mathfrak{p}}}$| which is unramified at |${\mathfrak{p}}$| equals |$L_{\mathfrak{p}}^{I_{\mathfrak{p}}}/k_{\mathfrak{p}}$|⁠, where |$L_{\mathfrak{p}}^{I_{\mathfrak{p}}}$| denotes the fixed field of |$I_{\mathfrak{p}}$| in |$L_{\mathfrak{p}}$|⁠; we denote it by |$L_{\mathfrak{p}}^{\textrm{{ur}}} /{k_{\mathfrak{p}}}$|⁠, and call it the unramified part of |${L_{\mathfrak{p}}} /{k_{\mathfrak{p}}}$|⁠. Furthermore, fixing a k-embedding σ of L into |$\widetilde{k_{\mathfrak{p}}}$| induces a choice of a prime |${\mathfrak{P}}$| lying over |${\mathfrak{p}}$| in L/k, via |${\mathfrak{P}}:=\left \{x\in L\, :\, v_{\mathfrak{p}}(\sigma (x))>0\right \}$|⁠, where |$v_{\mathfrak{p}}$| is the |${\mathfrak{p}}$|-adic valuation on |$k_{\mathfrak{p}}$|⁠, extended to |$\widetilde{k_{\mathfrak{p}}}$|⁠. 2.2 Extensions of function fields Let |$\widetilde k$| denote a fixed algebraic closure of a given field k of characteristic zero, and let T be an indeterminate over k. 2.2.1 Classical background Let E/k(T) be a finite Galois extension with Galois group G, and such that E/k is regular, that is, such that |$E \cap \widetilde{k}=k$|⁠. We then say that E/k(T) is a k-regular G-extension. A point t0 in |$\mathbb{P}^{1}\left (\widetilde{k}\right )$| is a branch point of E/k(T) if the prime ideal |$(T-t_{0}) \, \widetilde{k}\left [T-t_{0}\right ]$| of |$\widetilde{k}\left [T-t_{0}\right ]$| is ramified in |$E \cdot \widetilde{k}/\widetilde{k}(T)$|⁠. Here, one should take the compositum |$E \cdot \widetilde{k}$| of E and |$\widetilde{k}$| in a fixed algebraic closure of k(T) containing k, and replace T − t0 by 1/T if |$t_{0}=\infty $|⁠. The extension E/k(T) has finitely many branch points, denoted by |$t_{1},\dots ,t_{r}$| (one has r = 0 if and only if G is trivial). For each |$i \in \{1,\dots ,r\}$|⁠, denote the decomposition group (resp., the inertia group) of E(ti)/k(ti)(T) at (a fixed prime of E(ti) lying over) |$(T-t_{i}) \, k(t_{i})\left [T-t_{i}\right ]$| by |$D_{t_{i}}$| (resp., by |$I_{t_{i}}$|⁠). Given |$t_{0} \in \mathbb{P}^{1}(k)$|⁠, the residue extension of E/k(T) at (T − t0) k[T − t0] is denoted by |${E}_{t_{0}}/k$|⁠, and called the specialization of E/k(T) at t0. It is a finite Galois extension whose Galois group is canonically isomorphic to |$D_{t_{0}}/I_{t_{0}}$|⁠, where |$D_{t_{0}}$| (resp., |$I_{t_{0}}$|⁠) denotes the decomposition group (resp., the inertia group) of E/k(T) at (T − t0) k[T − t0]. If k is the fraction field of a Dedekind domain R, we denote the decomposition group (resp., the inertia group) of the specialization |$E_{t_{0}}/k$| at a given non zero prime ideal |${\mathfrak{p}}$| of R such that the residue field |$\overline{k}_{\mathfrak{p}}$| is perfect by |$D_{t_{0},{\mathfrak{p}}}$| (resp., by |$I_{t_{0},{\mathfrak{p}}}$|⁠). Note that |$D_{t_{0},{\mathfrak{p}}}$| is canonically isomorphic to a subgroup of |$D_{t_{0}}/I_{t_{0}}$|⁠. We recall the following well-known lemma, cf. [29], which is a standard consequence of Krasner’s lemma and the compatibility between the Hilbert specialization property and the weak approximation property of |$\mathbb{P}^{1}$|⁠: Lemma 2.2. Assume k is a number field. Let |$\mathcal{S}$| be a finite set of primes of k, and P(T, Y ) ∈ k[T][Y ] a monic, separable polynomial with splitting field E over k(T). For each |${\mathfrak{p}} \in \mathcal{S}$|⁠, choose |$t_{\mathfrak{p}}$| in k such that |$P(t_{\mathfrak{p}},Y)$| is separable, and let |$(E_{t_{\mathfrak{p}}})_{\mathfrak{p}}/k_{\mathfrak{p}}$| be the completion of |$E_{t_{\mathfrak{p}}}/k$| at |${\mathfrak{p}}$|⁠. Then, there exist infinitely many t0 ∈ k such that |$E_{t_{0}}/k$| has Galois group G and its completion at |${\mathfrak{p}}$| is |$(E_{t_{\mathfrak{p}}})_{\mathfrak{p}}/k_{\mathfrak{p}}$| for each |${\mathfrak{p}} \in \mathcal{S}$|⁠. Moreover, one may require these specializations |$E_{t_{0}}/k$| to be pairwise linearly disjoint. 2.2.2 Laurent series fields and their algebraic extensions Given |$t_{0} \in \mathbb{P}^{1}(k)$|⁠, set |${\mathfrak{p}}:=(T-t_{0}) \, k\left [T-t_{0}\right ]$|⁠. The unramified part |$E_{\mathfrak{p}}^{\textrm{{ur}}}/k((T-t_{0}))$| of the completion |$E_{\mathfrak{p}}/k((T-t_{0}))$| of E/k(T) at |${\mathfrak{p}}$| is of the form |$E_{t_{0}}((T-t_{0})) / k((T-t_{0}))$|⁠; see [34, page 55]. Furthermore, one has |$E_{\mathfrak{p}} = E_{\mathfrak{p}}^{{\textrm{{ur}}}}\left (\sqrt [e]{\pi }\right )$| for some element |$\pi \in E_{\mathfrak{p}}^{{\textrm{{ur}}}}$| of |${\mathfrak{p}}$|-adic valuation 1. Since π = α(T − t0)(1 + β) for some |$\alpha \in E_{t_{0}}$| and |$\beta \in{\mathfrak{p}} E_{\mathfrak{p}}^{{\textrm{{ur}}}}$|⁠, and since 1 + β is an e-th power in |$E_{\mathfrak{p}}^{{\textrm{{ur}}}}$|⁠, we may replace π by α(T − t0), so that |$E_{\mathfrak{p}} = E_{\mathfrak{p}}^{{\textrm{{ur}}}}\left (\sqrt [e]{\alpha (T-t_{0})}\right )$|⁠. Moreover, since |$E_{\mathfrak{p}}$| is the splitting field of Xe − α(T − t0), the e-th roots of unity are contained in |$E_{\mathfrak{p}}$|⁠, and then even in |$E_{\mathfrak{p}} \cap \widetilde{k} = E_{t_{0}}$|⁠, giving the following lemma. Lemma 2.3. Let ti be a branch point of E/k(T) of ramification index ei. Then, the specialization |$E(t_{i})_{t_{i}}$| contains all ei-th roots of unity. Setting k′ := k(ti), E′ := E(ti), and |${\mathfrak{p}}_{i}:=(T-t_{i}) \, k^\prime [T-t_{i}]$|⁠, the completion |$E^\prime _{{\mathfrak{p}}_{i}}$| is then a solution to the embedding problem |$D_{t_{i}}\stackrel{\varphi }{\to } \textrm{Gal}({E^{\prime }}^{\textrm{{ur}}}_{{\mathfrak{p}}_{i}}/k^\prime ((T-t_{i})))$|⁠, where the image is identified with |$D_{t_{i}}/I_{t_{i}}$| and φ is the natural projection. Since |$E^\prime _{{\mathfrak{p}}_{i}}$| is a Kummer extension of |$E^\prime _{t_{i}}((T-t_{i}))$|⁠, the conjugation action of |$D_{t_{i}}$| on |$I_{t_{i}}$| is isomorphic to the action on the group of ei-th roots of unity in |$E^\prime _{t_{i}}$|⁠. Such an embedding problem is then called a Brauer embedding problem. See, for example, [27, Chapter IV, Section 7] for more details. Finally, we will make use of a theorem about the structure of algebraic power series, which was first stated over the integers by Eisenstein. For our purposes, let R be an integral domain with fraction field k. The general version below can be found, for example, in [3, Lemma 2.1]. Theorem 2.4. Let |$\alpha = \sum _{n=0}^{\infty } \alpha _{n} T^{n} \in k[[T]]$| be algebraic over k(T). Then, there exist r and s in R such that |$r\cdot (\alpha _{n} s^{n}) \in R$| for each n ≥ 0. In particular, if R is a Dedekind domain, there exist only finitely many prime ideals |${\mathfrak{p}}$| of R such that α is not in |$R_{({\mathfrak{p}})}[[T]]$|⁠. 3 Ramification in Specializations In this section, we recall some classical properties on ramification in specializations of function field extensions. Let k be the fraction field of a Dedekind domain R of characteristic zero, and let |${\mathfrak{p}}$| be a non zero prime ideal of R such that |$\overline{k}_{\mathfrak{p}}$| is perfect. Let |$v_{\mathfrak{p}}$| denote the discrete valuation on k with valuation ring |$R_{({\mathfrak{p}})}$|⁠. Let T be an indeterminate over k and G a finite group. First, we recall the definition of meeting modulo |${\mathfrak{p}}$|: Definition 3.1. (1) Let F/k be a finite extension, RF the integral closure of R in F, and |${\mathfrak{p}}_{F}$| a non zero prime ideal of RF. We say that two distinct points t0 and t1 in |$\mathbb{P}^{1}(F)$|meet modulo |${\mathfrak{p}}_{F}$| if either |$v_{{\mathfrak{p}}_{F}}(t_{0}) \geq 0$|⁠, |$v_{{\mathfrak{p}}_{F}}(t_{1}) \geq 0$|⁠, and |$v_{{\mathfrak{p}}_{F}}(t_{0}-t_{1})> 0$|⁠, or |$v_{{\mathfrak{p}}_{F}}(t_{0}) \leq 0$|⁠, |$v_{{\mathfrak{p}}_{F}}(t_{1}) \leq 0$|⁠, and |$v_{{\mathfrak{p}}_{F}}((1/t_{0}) - (1/t_{1}))> 0$| (where |$v_{{\mathfrak{p}}_{F}}$| denotes the discrete valuation on F associated with |${\mathfrak{p}}_{F}$|⁠). Here, we set |$1/\infty = 0$|⁠, |$1 / 0 = \infty $|⁠, |$v_{{\mathfrak{p}}}(\infty ) = -\infty $|⁠, and |$v_{{\mathfrak{p}}}(0) = \infty $|⁠. (2) We say that two distinct points t0 and t1 in |$\mathbb{P}^{1}\left (\widetilde{k}\right )$|meet modulo |${\mathfrak{p}}$| if t0 and t1 meet modulo a prime ideal of the integral closure of R in k(t0, t1) lying over |${\mathfrak{p}}$|⁠. Note that, if t0 and t1 meet modulo |${\mathfrak{p}}$| and F/k is a finite extension containing k(t0, t1), then, t0 and t1 meet modulo a prime ideal of the integral closure of R in F lying over |${\mathfrak{p}}$|⁠. In the case where t0 is k-rational and meets t1 modulo |${\mathfrak{p}}$|⁠, the following lemma asserts the existence of a unique degree 1 prime lying over |${\mathfrak{p}}$| at which t0 and t1 meet. Lemma 3.2. For every |$t_{1}\in \mathbb{P}^{1}(\widetilde{k})$|⁠, there exists a finite set |$\mathcal S_{1}$| of primes of k, depending only on t1, which satisfies the following property. Suppose |${\mathfrak{p}}\not \in \mathcal S_{1}$| and let |$t_{0}\in \mathbb P^{1}(k)\setminus \{t_{1}\}$| be such that t0 and t1 meet modulo |${\mathfrak{p}}$|⁠. Then, there exists a unique prime |${\mathfrak{p}}^\prime :={\mathfrak{p}}^\prime \left (t_{0},t_{1},{\mathfrak{p}}\right )$| lying over |${\mathfrak{p}}$| in k(t1)/k with residue degree |$f_{{\mathfrak{p}}^\prime \vert{\mathfrak{p}}}=1$| at which t0 and t1 meet. Remark 3.3. (1) If t1 is k-rational, then, one has |${\mathfrak{p}}^\prime (t_{0},t_{1},{\mathfrak{p}})={\mathfrak{p}}$|⁠. (2) For each prime ideal |${\mathfrak{p}}^\prime $| lying over |${\mathfrak{p}} \not \in \mathcal{S}_{1}$| in k(t1)/k with residue degree |$f_{{\mathfrak{p}}^\prime | {\mathfrak{p}}}=1$|⁠, the weak approximation property of |$\mathbb{P}^{1}$| provides infinitely many t0 ∈ k such that |${\mathfrak{p}}^\prime (t_{0},t_{1},{\mathfrak{p}})={\mathfrak{p}}^\prime $|⁠. Proof. By part (1) of Remark 3.3, we may assume |$t_{1} \ne \infty $|⁠. We require the set |$\mathcal S_{1}$| to contain the (finite) set of primes |$\mathfrak{q}$| of k such that the minimal polynomial |$m_{t_{1}}(T)$| of t1 over k is either not integral at |$\mathfrak{q}$| or not separable modulo |$\mathfrak{q}$|⁠. In particular, for every prime |${\mathfrak{p}}^\prime $| of k(t1) lying over |${\mathfrak{p}}$|⁠, we may assume |$v_{{\mathfrak{p}}^\prime }(t_{1})\geq 0$| and |${\overline{k(t_{1})}}_{{\mathfrak{p}}^\prime }$| is generated over |${\overline{k}}_{\mathfrak{p}}$| by the reduction modulo |${\mathfrak{p}}^\prime $| of t1, by Lemma 2.1. For the existence part, note that, since t0 and t1 meet modulo |${\mathfrak{p}}$|⁠, the above provides a prime ideal |${\mathfrak{p}}^\prime $| lying over |${\mathfrak{p}}$| in k(t1)/k such that |$v_{{\mathfrak{p}}^\prime }(t_{0}) \geq 0$|⁠, |$v_{{\mathfrak{p}}^\prime }(t_{1}) \geq 0$|⁠, and |$v_{{\mathfrak{p}}^\prime }(t_{0}-t_{1})> 0$|⁠. Since |${\mathfrak{p}}$| is not in |$\mathcal{S}_{1}$|⁠, and since the reduction modulo |${\mathfrak{p}}^\prime $| of t1 equals the reduction |$\overline{t_{0}}$| modulo |${\mathfrak{p}}^\prime $| of t0, we have |${\overline{k(t_{1})}}_{{\mathfrak{p}}^\prime } = {\overline{k}}_{\mathfrak{p}}({\overline{t_{0}}}) = {\overline{k}}_{\mathfrak{p}}$|⁠, that is, |$f_{{\mathfrak{p}}^\prime | {\mathfrak{p}}}=1$|⁠. For the uniqueness part, assume that t0 and t1 meet modulo two distinct primes |${\mathfrak{p}}^\prime _{1}$| and |${\mathfrak{p}}^\prime _{2}$| of k(t1) lying over |${\mathfrak{p}}$|⁠, both with residue degree 1. Since |$m_{t_{1}}(T)$| is separable modulo |${\mathfrak{p}}$|⁠, the primes |${\mathfrak{p}}^\prime _{1}$| and |${\mathfrak{p}}^\prime _{2}$| correspond to distinct linear factors T − a1 and T − a2 of the reduction of |$m_{t_{1}}(T)$| in |${\overline{k}}_{\mathfrak{p}}[T]$|⁠, so that t1 ≡ aj modulo |${\mathfrak{p}}^\prime _{j}$| for each j ∈ {1, 2}. As t0 and t1 meet modulo |${\mathfrak{p}}^\prime _{j}$|⁠, the difference t0 − t1 is in |${\mathfrak{p}}^\prime _{j}$|⁠. Hence, t0 − aj is in |${\mathfrak{p}}^\prime _{j} \cap k= {\mathfrak{p}}$|⁠. We then get a1 ≡ a2 modulo |${\mathfrak{p}}$|⁠, which cannot happen. Now, we recall the definition of intersection multiplicity at |${\mathfrak{p}}$|. Below, the minimal polynomial over k of any given element |$t_{1} \in \mathbb{P}^{1}\left (\widetilde{k}\right )$| is denoted by |$m_{t_{1}}(T)$| (we set |$m_{t_{1}}(T)=1$| if |$t_{1} = \infty $|⁠). Denote the constant coefficient of |$m_{t_{1}}(T)$| by |$a_{t_{1}}$|⁠. Then, the minimal polynomial of 1/t1 over k is |$m_{1/t_{1}}(T)=(1/a_{t_{1}})\, T^{{\textrm{{deg}}}(m_{t_{1}}(T))} \, m_{t_{1}}(1/T)$| if |$t_{1} \in \widetilde{k} \setminus \{0\}$|⁠, |$m_{1/t_{1}}(T)=1$| if t1 = 0, |$m_{1/t_{1}}(T)=T$| if |$t_{1} = \infty $|⁠. Let t1 be in |$\mathbb{P}^{1}\left (\widetilde{k}\right )$| and t0 in |$\mathbb{P}^{1}(k)$|⁠. Assume |$v_{{\mathfrak{p}}}\left (a_{t_{1}}\right )=0$| if t1≠0 to make the intersection multiplicity well-defined in Definition 3.4 below. Definition 3.4. The intersection multiplicity |$I_{{\mathfrak{p}}}(t_{0},t_{1})$| of t0 and t1 at |${\mathfrak{p}}$| is $$ I_{{\mathfrak{p}}}(t_{0},t_{1})= \begin{cases} v_{{\mathfrak{p}}}(m_{t_{1}}(t_{0})) &\textrm{if}\ v_{{\mathfrak{p}}}(t_{0}) \geq 0, \\ v_{{\mathfrak{p}}}(m_{1/t_{1}}(1/t_{0})) &\textrm{if}\ v_{{\mathfrak{p}}}(t_{0}) \leq 0.\end{cases}$$ Lemma 3.5 below, which is [25, Lemma 2.5], connects Definitions 3.1 and 3.4. Lemma 3.5. Let t1 be in |$\mathbb{P}^{1}\left (\widetilde{k}\right )$| and t0 in |$\mathbb{P}^{1}(k)$|⁠. Assume |$v_{{\mathfrak{p}}}\left (a_{t_{1}}\right )=0$| if t1≠0. (1) If |$I_{{\mathfrak{p}}}(t_{0},t_{1})>0$|⁠, then, t0 and t1 meet modulo |${\mathfrak{p}}$|⁠. (2) The converse in (1) holds, provided |$m_{t_{1}}(T)$| is in |$R_{({\mathfrak{p}})}[T]$|⁠. Finally, we recall the following classical result on ramification in specializations; see, for example, [4, Proposition 4.2], [7], and [25, Section 2.2.3]. Let E/k(T) be a k-regular G-extension with branch points |$t_{1},\dots ,t_{r}$|⁠. For every |$i \in \{1, \dots ,r\}$|⁠, denote the inertia group of E(ti)/k(ti)(T) at (T − ti) k(ti)[T − ti] by |$I_{t_{i}}$|⁠. Specialization Inertia Theorem.Assume that|${\mathfrak{p}}$|is not in some finite set|$\mathcal{S}_{\textrm{{bad}}}:= \mathcal{S}_{\textrm{{bad}}}(E/k(T))$|of prime ideals of R (described explicitly in, e.g., [25, Section 2.2.3]). Let|$t_{0} \in \mathbb{P}^{1}(k) \setminus \{t_{1},\dots ,t_{r}\}$|⁠. (1) If|${\mathfrak{p}}$|ramifies in|$E_{t_{0}}/k$|⁠, then, t0meets some branch point tiof E/k(T) modulo|${\mathfrak{p}}$|⁠. (2) Suppose that t0and timeet modulo|${\mathfrak{p}}$|⁠. Then, |$I_{t_{0},{\mathfrak{p}}}$|is conjugate to|$I_{t_{i}}^{I_{{\mathfrak{p}}}(t_{0},t_{i})}$|⁠. 4 Main Results In this section, we state and prove Theorems 4.1, 4.4, and 4.6, which are the most general versions of Theorems 1.1, 1.2, and 1.3, respectively. Throughout this section, we use the notation of Sections 2–3. Let k be the fraction field of a Dedekind domain R of characteristic zero such that, for every prime ideal |${\mathfrak{p}}$| of R, the residue field |$\overline{k}_{\mathfrak{p}}$| is perfect. Given an indeterminate T over k and a finite group G, let E/k(T) be a k-regular G-extension with branch points |$t_{1},\dots ,t_{r}$|⁠. Fix i ∈ {1, …, r}. Recall that |$D_{t_{i}}$| (resp., |$I_{t_{i}}$|⁠) is the decomposition group (resp., the inertia group) of E(ti)/k(ti)(T) at (T − ti) k[T − ti], and that |$D_{t_{i},{\mathfrak{p}}^\prime }$| is the decomposition group of |$(E(t_{i}))_{t_{i}}/k(t_{i})$| at a prime |${\mathfrak{p}}^\prime $| of k(ti), so that |$D_{t_{i},{\mathfrak{p}}^\prime }$| is canonically identified with a subgroup of |$D_{t_{i}}/I_{t_{i}}$|⁠. Set |$e_{i}:=|I_{t_{i}}|$|⁠, and let |$\varphi :D_{t_{i}} \rightarrow D_{t_{i}}/I_{t_{i}}$| be the natural projection. 4.1 Statement of Theorem 4.1 Theorem 4.1. Assume that |$t_{0}\in \mathbb{P}^{1}(k) \setminus \{t_{1},\dots ,t_{r}\}$| and ti meet modulo a prime |${\mathfrak{p}}$| of k avoiding a finite set |$\mathcal{S}_{\textrm{{exc}}} := \mathcal{S}_{\textrm{{exc}}}(E/k(T))$| of primes of k. Let |${\mathfrak{p}}^\prime ={\mathfrak{p}}^\prime (t_{0},t_{i},{\mathfrak{p}})$| be the unique prime ideal lying over |${\mathfrak{p}}$| in k(ti)/k provided by Lemma 3.2 (to make this well-defined, we assume in particular that |$\mathcal{S}_{\textrm{{exc}}}$| contains the set |$\mathcal{S}_{1}$| from Lemma 3.2). (1) The decomposition group |$D_{t_{0},{\mathfrak{p}}}$| is conjugate in G to a subgroup U of |$D_{t_{i}}$| such that |$\varphi (U) = D_{t_{i}, {\mathfrak{p}}^\prime }$|⁠. Moreover, if |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei, one has |$U=\varphi ^{-1}\left (D_{t_{i}, {\mathfrak{p}}^\prime }\right )$|⁠. (2) The unramified part of the completion |$E_{t_{0}}\cdot k_{\mathfrak{p}}$| of |$E_{t_{0}}$| at |${\mathfrak{p}}$| contains |$\left (E(t_{i})\right )_{t_{i}}\cdot k(t_{i})_{{\mathfrak{p}}^\prime }$|⁠, with equality if |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei. As the specialization |$E_{t_{0}}/k$| is Galois, the compositum |$E_{t_{0}}\cdot k_{\mathfrak{p}}$| is independent of the embedding of |$E_{t_{0}}$| into a given algebraic closure of |$k_{\mathfrak{p}}$|⁠. Similarly, |$\left (E(t_{i})\right )_{t_{i}}\cdot k(t_{i})_{{\mathfrak{p}}^\prime }$| is independent of the embedding of |$(E(t_{i}))_{t_{i}}$| into a given algebraic closure of |$k(t_{i})_{{\mathfrak{p}}^\prime }$|⁠. Remark 4.2. (1) Let |${\mathfrak{p}}$| be a prime of k, not in |$\mathcal{S}_{\textrm{{exc}}}$|⁠, and let |${\mathfrak{p}}^\prime $| be a prime lying over |${\mathfrak{p}}$| in k(ti)/k with residue degree |$f_{{\mathfrak{p}}^\prime | {\mathfrak{p}}}=1$|⁠. As in part (2) of Remark 3.3, there exist infinitely many t0 ∈ k such that t0 and ti meet modulo |${\mathfrak{p}}^\prime $|⁠, and such that |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei. Thus, as a consequence of part (1) of Theorem 4.1 and of the Specialization Inertia Theorem, there exist infinitely many |$t_{0} \in k \setminus \{t_{1},\dots ,t_{r}\}$| such that |$I_{t_{0}, {\mathfrak{p}}}$| is conjugate to |$I_{t_{i}}$|⁠, and |$D_{t_{0},{\mathfrak{p}}}$| is conjugate to |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^\prime })$|⁠. (2) Part (2) of Theorem 4.1 implies that the residue degree of |$(E(t_{i}))_{t_{i}}/k(t_{i})$| at |${\mathfrak{p}}^\prime $| divides that of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$|⁠, with equality if |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei. (3) The assumption that |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei is necessary in general for the more precise conclusions in parts (1) and (2) of Theorem 4.1, as the following easy example shows. Set |$k:=\mathbb{Q}$|⁠, |$E:=\mathbb{Q}(\sqrt{T})$|⁠, ti := 0, and let p be a prime number. Since |$E/\mathbb{Q}(T)$| is totally ramified at ti, one has |$E_{t_{i}}=\mathbb{Q}$|⁠, and |$|I_{t_{i}}|=|D_{t_{i}}|=2$|⁠. Set t0 := α ⋅ p2, where α denotes a rational number of p-adic valuation 0. Then, one has |$({E_{t_{0}}})_{p}=\mathbb{Q}_{p}$| if α is a square modulo p (and hence |$1=|D_{t_{0},p}|<|\varphi ^{-1} (D_{t_{i},p})|=2$|⁠), whereas |$(E_{t_{0}})_{p}/\mathbb{Q}_{p}$| has degree 2 if α is not a square modulo p, and therefore |$E_{t_{i}}\cdot \mathbb{Q}_{p} \subsetneq E_{t_{0}}\cdot \mathbb{Q}_{p}$| in this case. 4.2 Proof of Theorem 4.1 By possibly changing the variable T, we may assume that ti is not equal to |$\infty $|⁠, and that ti is integral over R. For simplicity, set k′ := k(ti), |$k^\prime _{{\mathfrak{p}}^\prime } :=k(t_{i})_{{\mathfrak{p}}^\prime }$|⁠, E′ := E(ti), |$E^\prime _{t_{i}}:=(E(t_{i}))_{t_{i}}$|⁠, S := T − ti, and let R′ be the integral closure of R in k′. From now on, we fix an embedding of E into a given algebraic closure |$\widetilde{k^\prime _{{\mathfrak{p}}^\prime }((S))}$| of |$k^\prime _{{\mathfrak{p}}^\prime }((S))$|⁠. Every compositum of fields below has to be understood inside |$\widetilde{k^\prime _{{\mathfrak{p}}^\prime }((S))}$|⁠. We break the proof into four steps. 4.2.1 Step I Here, we identify the Galois group of E′⋅ k′((S))/k′((S)) with |$D_{t_{i}}$| and that of |$E^\prime \cdot k^\prime _{{\mathfrak{p}}^\prime }((S))/k^\prime _{{\mathfrak{p}}^\prime }((S))$| with a subgroup of |$D_{t_{i}}$|⁠, namely, with |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^\prime })$|⁠. First, consider the compositum of E′ and k′((S)). The restriction of the Galois group Gal(E′⋅ k′((S))/k′((S))) to E′/k′(S) preserves a prime |$\mathfrak{Q}$| of E′ lying over the prime generated by S. Hence, we identify Gal(E′⋅ k′((S))/k′((S))) with the decomposition group |$D_{t_{i}}$| of E′/k′(S) at |$\mathfrak{Q}$|⁠. Moreover, |$E^\prime _{t_{i}}/k^\prime $| is the residue extension of E′/k′(S) at |$\mathfrak{Q}$|⁠, and the unramified part of E′⋅ k′((S))/k′((S)) is |$E^\prime _{t_{i}} \cdot k^\prime ((S))/ k^\prime ((S))$|⁠; see Section 2.2.2. Thus, |$E^\prime \cdot k^\prime ((S))/ E^\prime _{t_{i}} \cdot k^\prime ((S))$| is totally ramified at the prime generated by S, and its Galois group is identified with the inertia group |$I_{t_{i}}$| of E′/k′(S) at |$\mathfrak{Q}$|⁠. Now, consider the compositum |$E^\prime _{t_{i}}\cdot k^\prime _{{\mathfrak{p}}^\prime }$|⁠. This defines an embedding of |$E^\prime _{t_{i}}$| into the algebraic closure of |$k^\prime _{{\mathfrak{p}}^\prime }$| that is contained in |$\widetilde{k^\prime _{{\mathfrak{p}}^\prime }((S))}$|⁠, and hence a prime |${\mathfrak{P}}^\prime $| of |$E^\prime _{t_{i}}$| lying over |${\mathfrak{p}}^\prime $| such that the restriction of |$\textrm{Gal}(E^\prime _{t_{i}}\cdot k^\prime _{{\mathfrak{p}}^\prime }/k^\prime _{{\mathfrak{p}}^\prime })$| to |$E^\prime _{t_{i}}/k^\prime $| preserves |${\mathfrak{P}}^\prime $|⁠. Therefore, we identify |$\textrm{Gal}(E^\prime _{t_{i}}\cdot k^\prime _{{\mathfrak{p}}^\prime }/k^\prime _{{\mathfrak{p}}^\prime })$| with the decomposition group |$D_{t_{i},{\mathfrak{p}}^\prime }$| of |$E^\prime _{t_{i}}/k^\prime $| at |${\mathfrak{P}}^\prime $|⁠. Next, consider the compositum |$E^\prime \cdot k^\prime _{{\mathfrak{p}}^\prime }((S))$|⁠. Its Galois group V over |$k^\prime _{{\mathfrak{p}}^\prime }((S))$| is identified with a subgroup of |$D_{t_{i}} = \textrm{Gal}(E^\prime \cdot k^\prime ((S))/k^\prime ((S)) )$| via restriction. Note that, as |$E^\prime \cdot k^\prime ((S)) / E^\prime _{t_{i}}\cdot k^\prime ((S))$| is totally ramified, the fields E′⋅ k′((S)) and |$E^\prime _{t_{i}}\cdot k^\prime _{{\mathfrak{p}}^\prime }((S))$| are linearly disjoint over |$E^\prime _{t_{i}}\cdot k^\prime ((S))$|⁠. Hence, |$\textrm{Gal}(E^\prime \cdot k^\prime _{{\mathfrak{p}}^\prime }((S))/E^\prime _{t_{i}}\cdot k^\prime _{{\mathfrak{p}}^\prime }((S)))$| is identified with |$I_{t_{i}} = \textrm{Gal}(E^\prime \cdot k^\prime ((S))/E^\prime _{t_{i}}\cdot k^\prime ((S)))$|⁠, so that |$\varphi (V) = D_{t_{i},{\mathfrak{p}}^\prime }$|⁠. We then obtain the following diagram of inclusions and Galois groups: 4.2.2 Step II Our reduction process modulo (T − t0) will occur in the domain |$R^\prime _{{\mathfrak{p}}^\prime }[[S]]$| (this domain is the completion of R′[S] at the maximal ideal generated by |${\mathfrak{p}}^\prime $| and S; see, for example, [13, Exercise 7.11].); see Step III (Section 4.2.3). Let F be the fraction field of |$R^\prime _{{\mathfrak{p}}^\prime }[[S]]$|⁠; note that |$F\subseteq k^\prime _{{\mathfrak{p}}^\prime }((S))$|⁠. Here, we show that |$E^\prime _{t_{i}}\cdot F$| is contained in E′⋅ F, and identify the Galois groups of |$E^\prime _{t_{i}}\cdot F/F$| and E′⋅ F/F with |$D_{t_{i},{\mathfrak{p}}^\prime }$| and |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^\prime } )$|⁠, respectively. This description rests on the following lemma: Lemma 4.3. For every finite extension M/k′(S), there exists a finite set |$\mathcal{S}_{M}$| of primes of k′ (depending only on M/k′(S)) such that, if |${\mathfrak{p}}^\prime $| is not in |$\mathcal{S}_{M}$|⁠, then, the fields M ⋅ F and |$k^\prime _{{\mathfrak{p}}^\prime }((S))$| are linearly disjoint over F. Proof. Up to replacing M/k′(S) by its Galois closure, we may assume that M/k′(S) is Galois. Given an intermediate field k′(S) ⊆ M0 ⊆ M, let α := α(M0) be a primitive element of M0/k′(S). We have \begin{align} \alpha=\sum_{i \geq -n_{0}} \alpha_{i} \cdot S^{i/e} \end{align} (4.2) for some positive integer e, some non negative integer n0, and coefficients αi, i ≥ −n0, which lie in a finite extension K/k′. By possibly replacing α by |$S^{n_{0}} \alpha $|⁠, we may assume n0 = 0. As α is algebraic over k′(S), Theorem 2.4 asserts that the set |$\mathcal T_{M_{0}}$| of all primes |${\mathfrak{q}}^{\prime}$| of k′ such that |$v_{{\mathfrak{Q}}^{\prime}}(\alpha _{j}) <0$| for some j ≥ 0 and some prime |${\mathfrak{Q}}^{\prime}$| of K lying over |${\mathfrak{q}}^{\prime}$| is finite, and depends only on α, that is, only on M0. Set |$\mathcal{S}_{M} := \cup _{M_{0}} \mathcal T_{M_{0}}$|⁠, where M0 runs over all intermediate fields in M/k′(S). Suppose that |${\mathfrak{p}}^{\prime}$| is not in |$\mathcal{S}_{M}$|⁠, and set |$F_{1}:=(M \cdot F)\,\cap \, k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$|⁠. To prove the lemma, it suffices to show that F1 = F. Since F ⊆ F1 ⊆ M ⋅ F, and since the latter is Galois over F, the field M1 := F1 ∩ M satisfies M1 ⋅ F = F1. Set α := α(M1). As α is an element of |$k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$|⁠, one has e = 1 (with the notation of (4.2)), so that α is an element of |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$| (as |${\mathfrak{p}}^{\prime}$| is not in |$\mathcal{S}_{M}$|⁠). Thus, α, and hence M1, are contained in F. Hence, F1, which is equal to M1 ⋅ F, is equal to F, thus ending the proof of the lemma. Apply Lemma 4.3 with |$M=E^{\prime} \cdot E^{\prime}_{t_{i}}$| to get a finite set |$\mathcal{S}_{M}$| of primes of k′, depending only on E/k(T), such that |$(E^{\prime} \cdot E^{\prime}_{t_{i}}) \cdot F$| and |$k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$| are linearly disjoint over F, provided |${\mathfrak{p}}^{\prime}$| is not in |$\mathcal{S}_{M}$|⁠. Set |$L:= (E^{\prime}_{t_{i}} \cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))) \cap (E^{\prime} \cdot F)$|⁠. Since |$(E^{\prime}_{t_{i}}\cdot F)\cdot k^{\prime}_{\mathfrak{p}}((S))\subseteq (E^{\prime}\cdot F)\cdot k^{\prime}_{\mathfrak{p}}((S))$|⁠, the previous linear disjointness provides |$E^{\prime}_{t_{i}}\cdot F\subseteq E^{\prime}\cdot F$|⁠. Hence, |$E^{\prime}_{t_{i}}\cdot F \subseteq L$|⁠. Conversely, L contains |$E^{\prime}_{t_{i}}\cdot F$| by its definition and the inclusion |$E^{\prime}_{t_{i}}\cdot F\subseteq E^{\prime}\cdot F$|⁠. Letting |$\mathcal{S}_{2}$| denote the finite set of primes of k obtained by restricting a prime in |$\mathcal{S}_{M}$|⁠, the equality |$L= E^{\prime}_{t_{i}}\cdot F$| holds, provided |${\mathfrak{p}} \not \in \mathcal{S}_{2}$|⁠. Set |$d_{i,{\mathfrak{p}}^{\prime}}:=[E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}:k^{\prime}_{{\mathfrak{p}}^{\prime}}]$|⁠. As F contains |$k^{\prime}_{{\mathfrak{p}}^{\prime}}=\textrm{Frac}(R^{\prime}_{{\mathfrak{p}}^{\prime}})$|⁠, $$ d_{i,{\mathfrak{p}}^{\prime}}=\left[E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}:k^{\prime}_{{\mathfrak{p}}^{\prime}}\right]\geq \left[E^{\prime}_{t_{i}}\cdot F:F\right] \geq \left[E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S)):k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))\right] = d_{i,{\mathfrak{p}}^{\prime}},$$ the last equality following from |$E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))/k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$| being unramified. Hence, |$[E^{\prime}_{t_{i}}\cdot F:F] = d_{i,{\mathfrak{p}}^{\prime}}$|⁠. Moreover, as E′⋅ F and |$k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$| are linearly disjoint over F, the Galois group |$\textrm{Gal} (E^{\prime}\cdot F/F)$| is identified with V (=|${\textrm{{Gal}}}(E^{\prime} \cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))/ k^{\prime}_{{\mathfrak{p}}^{\prime}}((S)))$|⁠; see (4.1)) via restriction. This identification gives |$\textrm{Gal}(E^{\prime}\cdot F/E^{\prime}_{t_{i}}\cdot F) = I_{t_{i}}$|⁠, and |$\textrm{Gal}(E^{\prime}_{t_{i}}\cdot F/F) = D_{t_{i},{\mathfrak{p}}^{\prime}}$|⁠. We then obtain the following diagram of inclusions and Galois groups: 4.2.3 Step III Here, we reduce the extension E′⋅ F/F modulo (T − t0), and identify the reduction with |$E_{t_{0}}k_{{\mathfrak{p}}}/k_{{\mathfrak{p}}}$|⁠. More precisely, we show that there exists a place |$\mathfrak{M}$| of E′⋅ F whose restriction to E has residue field |$E_{t_{0}}$| and whose restriction to F has residue field |$k_{\mathfrak{p}}$|⁠. We then show that the residue field at |$\mathfrak{M}$| is the compositum |$E_{t_{0}} k_{\mathfrak{p}}$|⁠. As t0 and ti meet modulo |${\mathfrak{p}}^{\prime}$| by the definition of |${\mathfrak{p}}^{\prime}$|⁠, and as ti is integral over R, one has |$v_{{\mathfrak{p}}^{\prime}}(t_{0}-t_{i})>0$|⁠. Hence, T − t0 = S − (t0 − ti) is in |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$|⁠. Moreover, as |$ ((t_{0}-t_{i})^{m})_{m \geq 1}$| converges to 0 in |$R^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠, the specialization map |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]\rightarrow R^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠, which sends S to t0 − ti, is well-defined. As it is on to, there is a canonical isomorphism |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]/(T-t_{0}) \, R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]\cong R^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠. In particular, |$(T-t_{0}) \, R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$| is a prime ideal of |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$|⁠. Let |$\mathfrak{R}$| be the localization of |$R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$| at |$(T-t_{0})\, R^{\prime}_{{\mathfrak{p}}^{\prime}}[[S]]$|⁠. The previous isomorphism shows that the residue field of |$\mathfrak{R}$| at |$(T-t_{0})\, \mathfrak{R}$| is canonically isomorphic to |${\textrm{{Frac}}}(R^{\prime}_{{\mathfrak{p}}^{\prime}}) = k^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠. Let |$\mathcal{S}_{3}$| be the finite set of primes of k which ramify in k′/k. Assuming |${\mathfrak{p}} \not \in \mathcal{S}_{3}$|⁠, and using that |$f_{{\mathfrak{p}}^{\prime} | {\mathfrak{p}}}=1$| by the definition of |${\mathfrak{p}}^{\prime}$|⁠, we get |$k^{\prime}_{{\mathfrak{p}}^{\prime}} = k_{\mathfrak{p}}$|⁠. Hence, |$\mathfrak{R}/(T-t_{0})\, \mathfrak{R}$| is canonically isomorphic to |$k_{\mathfrak{p}}$|⁠. We use this canonical isomorphism to identify the two fields. The integral closure |$\mathfrak{S}$| of |$\mathfrak{R}$| in E′⋅ F is a Dedekind domain containing a prime ideal |$\mathfrak{M}$| lying over |$(T-t_{0}) \, \mathfrak{R}$|⁠. In particular, |$\mathfrak{M}\cap E$| is a prime of E lying over (T − t0) k[T]. Let |$\mathfrak R_{t_{i}}$| (resp., |$\mathfrak{M}_{t_{i}}$|⁠) be the restriction to |$E^{\prime}_{t_{i}}\cdot F$| of |$\mathfrak{S}$| (resp., of |$\mathfrak{M}$|⁠). Since |$E^{\prime}_{t_{i}}$| and |$k^{\prime}_{{\mathfrak{p}}^{\prime}}$| are contained in the residue field |$\mathfrak{R}_{t_{i}}/\mathfrak{M}_{t_{i}}$|⁠, and since |$[E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}:k^{\prime}_{{\mathfrak{p}}^{\prime}}]=d_{i,{\mathfrak{p}}^{\prime}} = [E^{\prime}_{t_{i}}\cdot F:F]$|⁠, we have |$\mathfrak{R}_{t_{i}}/\mathfrak{M}_{t_{i}}=E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠. Thus, we have the following diagram of inclusions: Now, we claim that |$\mathfrak{S}/\mathfrak{M}$| is equal to the compositum of |$k_{\mathfrak{p}}$| and |$E_{t_{0}}$|⁠, provided |${\mathfrak{p}}$| is not in some finite set of primes of k defined below. Indeed, let P(T, Y ) ∈ R[T][Y ] be the minimal polynomial over k(T) of some primitive element of E/k(T), assumed to be integral over R[T]. If P(t0, Y ) is not separable, then, t0 belongs to the finite set D of all roots in k of the discriminant Δ(T) ∈ R[T] of P(T, Y ) which are not branch points of E/k(T). As t0 and ti meet modulo |${\mathfrak{p}}$|⁠, the inseparability of P(t0, Y) implies that |${\mathfrak{p}}$| is in the finite set |$\mathcal{S}_{4}$| of primes |${\mathfrak{q}}$| of k such that |$v_{{\mathfrak{q}}^{\prime}}(d-t_{i})>0$| for some d ∈ D and some prime |${\mathfrak{q}}^{\prime}$| of k′ lying over |${\mathfrak{q}}$|⁠. Henceforth, we shall assume that |${\mathfrak{p}}$| is not in |$\mathcal{S}_{4}$|⁠, and then that P(t0, Y ) is separable. Denote the roots of P(T, Y ) in E by y1, …, yn, and their reductions modulo |$\mathfrak{M}\cap E$| by |${\overline{y}}_{1},\dots ,{\overline{y}}_{n}$|⁠, respectively. As Δ(t0)≠0, the discriminant Δ(T) is not in |$(T-t_{0}) \, \mathfrak{R}$|⁠. Since E′⋅ F is generated by y1, …, yn over F, the residue field |$\mathfrak{S}/\mathfrak{M}$| is generated by |${\overline{y}}_{1},\ldots ,{\overline{y}}_{n}$| over |$\mathfrak{R}/(T-t_{0}) \, \mathfrak{R}=k_{\mathfrak{p}}$| by Lemma 2.1. Thus, one has |$\mathfrak{S}/\mathfrak{M} = E_{t_{0}}\cdot k_{\mathfrak{p}}$|⁠, proving the claim. The general containment in part (2) of Theorem 4.1 follows from (4.4), provided |${\mathfrak{p}}$| is not in |$\mathcal{S}_{1}\cup \mathcal{S}_{2}\cup \mathcal{S}_{3}\cup \mathcal{S}_{4}$|⁠. 4.2.4 Step IV Here, we use Step III to identify the Galois group of |$E_{t_{0}}k_{{\mathfrak{p}}}/k_{{\mathfrak{p}}}$| with a subgroup of |$\textrm{Gal}(E^{\prime}\cdot F/F)=\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^{\prime}} )$| whose projection under φ equals |$D_{t_{i},{\mathfrak{p}}^{\prime}}$|⁠. Assume that |${\mathfrak{p}}\not \in \mathcal{S}_{1}\cup \mathcal{S}_{2}\cup \mathcal{S}_{3}\cup \mathcal{S}_{4}$|⁠, so that we are in the situation of (4.3) and (4.4). Up to replacing |$D_{t_{0}}$| by a conjugate of it, we may assume that |$D_{t_{0}}$| is the decomposition group of E/k(T) at |$\mathfrak{M}\cap E$|⁠. Let U be the decomposition group of E′⋅ F/F at |$\mathfrak{M}$|⁠, so that it identifies via restriction with a subgroup of |$D_{t_{0}}$|⁠. As the prime of k[T] generated by T − t0 is unramified in E/k(T), it is also unramified in E′⋅ F/F. Thus, U (resp., |$D_{t_{0}}$|⁠) is also the Galois group of |$E_{t_{0}}\cdot k_{\mathfrak{p}}/k_{\mathfrak{p}}$| (resp., of |$E_{t_{0}}/k$|⁠). Hence, the restriction of U to |$E_{t_{0}}/k$| is the decomposition group of some prime of |$E_{t_{0}}$| lying over |${\mathfrak{p}}$|⁠. Thus, we may identify U with |$D_{t_{0},{\mathfrak{p}}}$|⁠. On the other hand, U is a subgroup of |$\textrm{Gal}(E^{\prime}\cdot F/F)$|⁠. The latter is identified with V in Step II (Section 4.2.2), and hence is a subgroup of |$D_{t_{i}}$| via the identification in Step I (Section 4.2.1). Moreover, φ(U) is the decomposition group of |$E^{\prime}_{t_{i}}\cdot F/F$| at |$\mathfrak{M}_{t_{i}}$|⁠. As shown in Step III (Section 4.2.3), one has |$\textrm{Gal}(E^{\prime}_{t_{i}}\cdot F/F) ={\textrm{{Gal}}}(E^{\prime}_{t_{i}} \cdot k_{{\mathfrak{p}}^{\prime}} / k_{{\mathfrak{p}}^{\prime}})$|⁠. But the latter is |$D_{t_{i},{\mathfrak{p}}^{\prime}}$| by Step II. Hence, |$\varphi (U) = D_{t_{i},{\mathfrak{p}}^{\prime}}$|⁠, thus proving the general case of part (1) of Theorem 4.1. Finally, assume that |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei. Let |$I_{\mathfrak{M}}$| denote the inertia group of E′⋅ F/F at |$\mathfrak{M}$|⁠. Since |$E^{\prime}_{t_{i}}\cdot F/F$| is unramified at |$\mathfrak{M}_{t_{i}}$|⁠, one has |$I_{\mathfrak{M}}\subseteq{\textrm{{Gal}}}(E^{\prime} \cdot F /E^{\prime}_{t_{i}} \cdot F)$| (⁠|$=I_{t_{i}}$|⁠; see Section 4.2.2). Set |$\mathcal{S}_{\textrm{{exc}}} := \mathcal{S}_{1}\cup \mathcal{S}_{2}\cup \mathcal{S}_{3}\cup \mathcal{S}_{4} \cup \mathcal{S}_{\textrm{{bad}}}$|⁠, where |$\mathcal{S}_{\textrm{{bad}}}$| is the finite set of primes of k from the Specialization Inertia Theorem (Section 3), and assume that |${\mathfrak{p}}$| is not in |$\mathcal{S}_{\textrm{{exc}}}$|⁠. As |$I_{\mathfrak{p}}(t_{0},t_{i})$| is coprime to ei, one has |$|I_{t_{0},{\mathfrak{p}}}|=e_{i}$| by the Specialization Inertia Theorem. Since the conjugation of U to |$D_{t_{0},{\mathfrak{p}}}$| sends |$I_{\mathfrak{M}}$| to |$I_{t_{0},{\mathfrak{p}}}$|⁠, we also have |$|I_{\mathfrak{M}}|=e_{i}$|⁠. As |$I_{\mathfrak{M}}$| is a subgroup of |$I_{t_{i}}$|⁠, and since |$|I_{t_{i}}|=e_{i}$|⁠, we get the equality |$I_{\mathfrak{M}} = I_{t_{i}}$|⁠. Thus, |$I_{t_{i}}$| is contained in U, and |$\mathfrak{M}_{t_{i}}$| is totally ramified in |$E^{\prime}\cdot F/E^{\prime}_{t_{i}}\cdot F$|⁠. Hence, |$U=\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^{\prime}})$|⁠, and |$\left (E_{t_{0}}\cdot k_{\mathfrak{p}}\right )^{\textrm{{ur}}} = \mathfrak{R}_{t_{i}}/\mathfrak{M}_{t_{i}} = E^{\prime}_{t_{i}}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}$|⁠, completing the proof of Theorem 4.1. 4.3 On specifying completions in specializations Theorem 4.1 restricts the structure of the completion |$(E_{t_{0}})_{\mathfrak{p}}$| at |${\mathfrak{p}}$| of |$E_{t_{0}}$|⁠. Namely, it implies that |$(E_{t_{0}})_{\mathfrak{p}}$| contains the field |$N^{({\mathfrak{p}})}:=\left (E(t_{i})\right )_{t_{i}}\cdot k(t_{i})_{{\mathfrak{p}}^{\prime}}$| whose Galois group over |$k(t_{i})_{{\mathfrak{p}}^{\prime}}$| is |$D_{t_{i},{\mathfrak{p}}^{\prime}}$|⁠, and that the Galois group |$\textrm{Gal}((E_{t_{0}})_{\mathfrak{p}}/k_{\mathfrak{p}})$| is a subgroup of |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^{\prime}})$|⁠, where |$\varphi :D_{t_{i}}\rightarrow D_{t_{i}}/I_{t_{i}}$| is the natural projection. The following theorem shows that this is the only restriction for extensions of |$k_{\mathfrak{p}}$| with Galois group |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^{\prime}})$|⁠: Theorem 4.4. Let |$\mathcal{S}$| be a finite set of primes of k, disjoint from some finite set |$\mathcal{S}^{\prime}_{\textrm{{exc}}}= \mathcal{S}^{\prime}_{\textrm{{exc}}}\left (E/k(T)\right )$|⁠. For each |${\mathfrak{p}}\in \mathcal{S}$|⁠, assume that there exists |$i:=i({\mathfrak{p}})\in \{1, \dots ,r\}$| and a prime |${\mathfrak{p}}^{\prime}$| lying over |${\mathfrak{p}}$| in k(ti)/k with residue degree 1, and let |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| be a finite Galois extension containing |$N^{({\mathfrak{p}})}:=(E(t_{i}))_{t_{i}}\cdot k(t_{i})_{{\mathfrak{p}}^{\prime}}$| such that there exists an isomorphism ψ from |${\textrm{{Gal}}} (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})$| to |$\varphi ^{-1}(D_{t_{i},{\mathfrak{p}}^{\prime}})$| which maps |${\textrm{{Gal}}}(L^{({\mathfrak{p}})}/N^{({\mathfrak{p}})})$| on to |$I_{t_{i}}$|⁠. Then, there exist infinitely many t0 ∈ k such that the completion of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| equals |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| for each |${\mathfrak{p}}\in \mathcal{S}$|⁠. Moreover, if k is a number field, we may assume that these specializations of E/k(T) have Galois group G. Remark 4.5. (1) If |$L^{({\mathfrak{p}})}/N^{({\mathfrak{p}})}$| is totally ramified, |${\mathfrak{p}} \in \mathcal{S}$|⁠, and if k is a number field, then, we are in the settings of Theorem 1.2. In this case, by the Specialization Inertia Theorem, the intersection multiplicity of t0 and ti at |${\mathfrak{p}}$| is coprime to |$|I_{t_{i}}|$|⁠, |${\mathfrak{p}} \in \mathcal{S}$|⁠. (2) By combining Theorem 4.4, [11, Theorem 1.2], and the Chinese remainder theorem, one can more generally require the extensions |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$|⁠, |${\mathfrak{p}} \in \mathcal{S}$|⁠, to be either of the above form or unramified of degree d, where d is an arbitrary positive integer such that G contains at least one element of order d. Proof. Fix a prime |${\mathfrak{p}}\in \mathcal{S}$|⁠. As in Section 4.2, set k′ := k(ti), |$k^{\prime}_{{\mathfrak{p}}^{\prime}} :=k(t_{i})_{{\mathfrak{p}}^{\prime}}$|⁠, E′ := E(ti), |$E^{\prime}_{t_{i}}:=(E(t_{i}))_{t_{i}}$|⁠, and S := T − ti. Moreover, set |$N:=N^{({\mathfrak{p}})}$|⁠, and |$\Gamma :=\varphi ^{-1} (D_{t_{i},{\mathfrak{p}}^{\prime}})$|⁠. As shown in Step I of the proof of Theorem 4.1 (Section 4.2.1), one has |${\textrm{{Gal}}} (N/k_{\mathfrak{p}})=D_{t_{i},{\mathfrak{p}}^{\prime}}$|⁠. Then, all fields |$L^{({\mathfrak{p}})}$| with the properties mentioned in Theorem 4.4 are solution fields to the embedding problem |$\Gamma \stackrel{\varphi }{\to }{\textrm{{Gal}}}(N/k_{\mathfrak{p}} )$|⁠. Since |$I_{t_{i}}$| (resp., Γ) is the inertia group (resp., the Galois group) of |$E^{\prime}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S)) / k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$| (as shown in Step I of the proof of Theorem 4.1 (Section 4.2.1)), the action of Γ on the cyclic kernel |$I_{t_{i}}$| is isomorphic to the action on ei-th roots of unity in N (see Section 2.2.2). The above embedding problem is then a Brauer embedding problem. By [27, Chapter IV, Theorem 7.2], if x ∈ N is chosen such that |$N\left (\sqrt [e_{i}]{x}\right )$| is a solution field to this embedding problem, then, all the solutions fields are of the form |$N\left (\sqrt [e_{i}]{\beta x}\right )$| with |$\beta \in k_{\mathfrak{p}}^{\times }$|⁠. Furthermore, upon multiplying β by a suitable ei-th power, we can require βx to be of positive |${\mathfrak{p}}$|-adic valuation. The field E′⋅ k′((S)) is generated over |$E^{\prime}_{t_{i}}\cdot k^{\prime}((S))$| by |$\sqrt [e_{i}]{\alpha S}$| for some |$\alpha \in E^{\prime}_{t_{i}}$|⁠; see Section 2.2.2. Up to enlarging the set |$\mathcal{S}^{\prime}_{\textrm{{exc}}}$|⁠, we may assume that α is of |${\mathfrak{p}}$|-adic valuation 0. Set |$M:=(E^{\prime}_{t_{i}}\cdot k^{\prime}(S) )(\sqrt [e_{i}]{\alpha S})$|⁠, and consider the field F from Step II of the proof of Theorem 4.1 (Section 4.2.2). Assume that |$\mathcal{S}^{\prime}_{\textrm{{exc}}}$| contains the set |$\mathcal{S}_{\textrm{{exc}}}$| from Theorem 4.1. Then, |$(E^{\prime}\cdot M)\cdot F$| and |$k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$| are linearly disjoint over F by Lemma 4.3. But one also has |$(M\cdot F) \cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S)) = E^{\prime}\cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}((S))$|⁠. Hence, M ⋅ F = E′⋅ F. Thus, we have |$E^{\prime}\cdot F = (E^{\prime}_{t_{i}}\cdot F)(\sqrt [e_{i}]{\alpha S})$|⁠. Next, we choose |$t_{0}\in k_{\mathfrak{p}}$| with |$I_{\mathfrak{p}}\left (t_{0},t_{i}\right )>0$| and specialize E′⋅ F/F at T − t0 as in Step III of the proof of Theorem 4.1 (Section 4.2.3). Since |$E^{\prime}_{t_{i}}\cdot F$| specializes to |$E^{\prime}_{t_{i}} \cdot k^{\prime}_{{\mathfrak{p}}^{\prime}}$| and |$\sqrt [e_{i}]{\alpha S}$| specializes to |$\sqrt [e_{i}]{\alpha (t_{0}-t_{i})}$|⁠, Lemma 2.1 yields that all fields of the form |$(E^{\prime}_{t_{i}} \cdot k^{\prime}_{{\mathfrak{p}}^{\prime}})(\sqrt [e_{i}]{\alpha \pi } )$||$(=N(\sqrt [e_{i}]{\alpha \pi }))$|⁠, with π an element of |$k_{\mathfrak{p}}=k^{\prime}_{{\mathfrak{p}}^{\prime}}$| of positive |${\mathfrak{p}}$|-adic valuation, can be reached via specializing T to |$t_{0}\in k_{\mathfrak{p}}$|⁠. Hence, if |$N\left (\sqrt [e_{i}]{x}\right )$| is one of the fields obtained in this way, then, we reach all fields of the form |$N\left (\sqrt [e_{i}]{\beta x}\right )$|⁠, where β is any element of |$k_{\mathfrak{p}}^{\times }$| such that βx is of positive |${\mathfrak{p}}$|-adic valuation. As shown above, this covers in particular the extension |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$|⁠. Krasner’s lemma shows that |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| can even be obtained by restricting to specialization values |$t_{\mathfrak{p}}\in k$|⁠. The extension |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| therefore equals |$(E_{t_{\mathfrak{p}}} )_{\mathfrak{p}}/k_{\mathfrak{p}}$|⁠, the latter being the specialization of |$E^{\prime}_{t_{i}}\cdot F/F$| at |$T-t_{\mathfrak{p}}$|⁠. Finally, choose t0 ∈ k sufficiently close |${\mathfrak{p}}$|-adically to every |$t_{\mathfrak{p}}\in \mathcal{S}$|⁠. This yields the assertion in the general case. In the number field case, Lemma 2.2 provides the conclusion on the Galois group of the produced specializations. 4.4 On specifying inertia and decomposition groups of specializations Our next result is devoted to a reoccurrence property for subgroups of G appearing as decomposition groups of specializations, in the case where k is a number field. Theorem 4.6. Assume that k is a number field, and let s be a positive integer. Given |$j \in \{1,\dots ,s\}$|⁠, choose a branch point ti(j) of E/k(T), and an element τi(j) of |$D_{t_{i(j)}}$|⁠. Then, there exist s infinite sets |$\mathcal{S}_{1}, \dots , \mathcal{S}_{s}$| of primes of k which satisfy the following property. For every tuple |$({\mathfrak{p}}_{1}, \dots , {\mathfrak{p}}_{s}) \in \mathcal{S}_{1} \times \dots \times \mathcal{S}_{s}$| of distinct primes, there exist infinitely many t0 in k for which |$E_{t_{0}}/k$| is a G-extension whose decomposition group (resp., inertia group) at |${\mathfrak{p}}_{j}$| is |$\langle I_{t_{i(j)}}, \tau _{i(j)} \rangle $| (resp., |$I_{t_{i(j)}}$|⁠) for j = 1, …, s. Proof. We may assume without loss that s = 1. Indeed, Lemma 2.2 yields that the assertion for one prime |${\mathfrak{p}}_{i}$| at a time (⁠|$i=1,\dots ,s$|⁠) implies the assertion for all primes |${\mathfrak{p}}_{1},\dots ,{\mathfrak{p}}_{s}$| simultaneously. Also, by that lemma, we do need to prove that |$E_{t_{0}}/k$| has Galois group G. As before, set i := i(1), ti := ti(1), τ := τi(1), k′ := k(ti), and |$E^{\prime}_{t_{i}}:= (E(t_{i}))_{t_{i}}$|⁠. Let |$\overline{\tau }$| be the image of τ under the natural projection |$D_{t_{i}} \rightarrow D_{t_{i}}/I_{t_{i}}$|⁠, and let C be the conjugacy class in |$D_{t_{i}}/I_{t_{i}}$| of |$\overline{\tau }$|⁠. By part (1) of Theorem 4.1 and part (1) of Remark 4.2, it suffices to show that there is an infinite set |$\mathcal{S}$| of primes of k such that, for every |${\mathfrak{p}} \in \mathcal{S}$|⁠, there is a prime |${\mathfrak{p}}^{\prime}$| lying over |${\mathfrak{p}}$| in k′/k, with residue degree |$f_{{\mathfrak{p}}^{\prime} | {\mathfrak{p}}}=1$|⁠, and such that the Frobenius|${\textrm{{Frob}}}_{{\mathfrak{p}}^{\prime}}(E^{\prime}_{t_{i}} / k^{\prime})$| lies in C. By the Chebotarev density theorem, the natural density of the set |$\mathcal{S^{\prime}}$| of all primes |${\mathfrak{p}}^{\prime}$| of k′ such that |${\textrm{{Frob}}}_{{\mathfrak{p}}^{\prime}} (E^{\prime}_{t_{i}} / k^{\prime})$| lies in C equals |$|C| \cdot |I_{t_{i}}| / |D_{t_{i}}|$|⁠. This remains true for the set |$\mathcal{S}^{\prime\prime}=\left \{{\mathfrak{p}}^{\prime}\in \mathcal{S}^{\prime} : f_{{\mathfrak{p}}^{\prime} | {\mathfrak{p}}^{\prime} \cap k}=1\right \}$|⁠, since the set of residue degree 1 primes |${\mathfrak{p}}^{\prime}$| of k′ is of natural density 1, by the prime number theorem for number fields. In particular, the set of all primes |${\mathfrak{p}}$| of k which are contained in a prime |${\mathfrak{p}}^{\prime} \in \mathcal{S}^{\prime\prime}$| is infinite, completing the proof. 5 Application to G-crossed Products and Admissibility This section is devoted to our application to G-crossed products over number fields. We state and prove our admissibility criterion, Theorem 5.2, in Section 5.2, and apply it to obtain explicit families of examples in Section 5.3. For this section, let k be a number field, R its ring of integers, T an indeterminate over k, and G a finite group. 5.1 Background Recall that G is called k-admissible if there exists a G-crossed product division algebra with center k. A G-extension L/k such that L is a maximal subfield of a G-crossed product division algebra is called k-adequate. To prove Theorem 5.2, we need Schacher’s admissibility criterion [33, Section 2] over number fields: Theorem 5.1. Let L/k be a G-extension. Then, L is k-adequate if and only if, for each prime number p dividing |G|, there exist two distinct prime ideals |${\mathfrak{p}}_{1}$| and |${\mathfrak{p}}_{2}$| of R such that the decomposition group of L/k at |${\mathfrak{p}}_{i}$| contains a p-Sylow subgroup of G for i = 1, 2. 5.2 A new general admissibility criterion Theorem 5.2. Assume that G has a k-regular realization E/k(T) such that, for every prime number p such that G has a non cyclic p-Sylow subgroup, the following holds: (H) for some branch point ti of E/k(T), the decomposition group |$D_{t_{i}}$| contains a p-Sylow subgroup P of G such that |$PI_{t_{i}}/I_{t_{i}}$| is cyclic. Then, there exist infinitely many pairwise linearly disjoint k-adequate G-extensions of k, arising as specializations of E/k(T). In particular, G is k-admissible. Proof. Let p be a prime number dividing |G|, and let P be a p-Sylow subgroup of G. First, assume that P is cyclic. Let tp ∈ k be such that |$E_{t_{p}}/k$| has Galois group G; such a tp exists by Hilbert’s irreducibility theorem. By the Chebotarev density theorem, there exist infinitely many prime ideals |${\mathfrak{p}}$| of R such that |$D_{t_{p},{\mathfrak{p}}}$| contains P. Now, assume that P is not cyclic. Let ti(p) be a branch point of E/k(T) such that the decomposition group |$D_{t_{i}(p)}$| contains P, and such that |$PI_{t_{i}(p)}/I_{t_{i}(p)}$| is cyclic (condition (H)). Then, by Theorem 4.6, there exists an infinite set |$\mathcal{S}_{p}$| of prime ideals of R such that, for each |${\mathfrak{p}} \in \mathcal{S}_{p}$|⁠, there exist infinitely many tp ∈ k such that |$D_{t_{p},{\mathfrak{p}}}=PI_{t_{i}(p)}$|⁠. By applying Lemma 2.2, we obtain t0 ∈ k such that |$E_{t_{0}}/k$| is a G-extension and, for each p dividing |G|, one has |$D_{t_{0},{\mathfrak{p}}}=PI_{t_{i}(p)}$| for at least two distinct primes |${\mathfrak{p}}$| of k, verifying Theorem 5.1. 5.3 Examples Roughly speaking, in order to apply Theorem 5.2, one has to produce k-regular realizations of G with “sufficiently large” ramification indices and residue degrees at some branch points. Ramification indices can often be prescribed purely group-theoretically (e.g., via the rigidity method). To ensure “large” residue degrees we use Lemma 2.3. This method yields a large class of new examples. Below, we derive the first |${\mathbb{Q}}$|-admissibility results for infinite families of finite non-abelian simple groups. More precisely, Theorem 5.3 covers half of the groups |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| (p prime), whereas Theorem 5.4 covers a further 1/8, giving a total of 62.5% of the groups. Theorem 5.3. Assume that k does not contain |$\sqrt{-1}$|⁠. Let p be a prime number such that either p ≡ 3 (mod 8) or p ≡ 5 (mod 8). Then, the groups |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| and |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$| are k-admissible. Proof. Recall that |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$| has order (p − 1)p(p + 1). Then, by our assumption on p, the 2-Sylow subgroups of |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$| (resp., of |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$|⁠) have order 8 (resp., 4). By [27, Chapter I, Corollary 8.10], there exists a k-regular |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$|-extension E/k(T) with k-rational branch points t1, t2, and t3, and such that the corresponding inertia groups |$I_{t_{1}}$|⁠, |$I_{t_{2}}$|⁠, and |$I_{t_{3}}$| are generated by elements in the conjugacy classes 2B, 4A, and pA of |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$|⁠, respectively (according to the ATLAS notation [5]). Consider the branch point t2 of E/k(T) with ramification index 4. By Lemma 2.3, and as k does not contain |$\sqrt{-1}$|⁠, the residue degree at this branch point is divisible by 2. It then remains to combine Theorem 5.2 and the classical fact that the Sylow subgroups of |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$| of odd order are cyclic to get that |${\textrm{{PGL}}}_{2}\left (\mathbb{F}_{p}\right )$| is k-admissible. Furthermore, the fixed field of |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| in E is a rational function field, say k(S), by, for example, [35, Lemma 4.5.1], that has degree 2 over k(T). Denote by s2 the unique point lying over t2 in k(S)/k(T). Then, s2 is a branch point of E/k(S) with ramification index 2, and the residue degree at s2 in E/k(S) is divisible by 2. As above, we conclude that |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| is k-admissible. Theorem 5.4. Let p be a prime number such that the following two conditions hold: (1) either p ≡ 7 (mod 16) or p ≡ 9 (mod 16), (2) either p ≡ 2 (mod 5) or p ≡ 3 (mod 5). Assume that k contains neither |$\sqrt{-1}$| nor |$\sqrt{p^{\star }}$|⁠, where p⋆ = (−1)(p−1)/2p. Then, the group |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| is k-admissible. Proof. By condition (1), the 2-Sylow subgroups of |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| have order 8. Moreover, 5 is not a square modulo p by condition (2). By [27, Chapter I, Theorem 7.10], one then has a |$\mathbb{Q}$|-regular |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$|-extension of |$\mathbb{Q}(T)$| with four branch points, and the corresponding inertia groups are generated by elements in the conjugacy classes 4A, 4A, pA, and pB of |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| (according to the ATLAS notation). The proof of that theorem also gives the position of the branch points of this |$\mathbb{Q}$|-regular extension. Namely, in the notation of that proof, the |$\mathbb{Q}$|-regular realization of |${\textrm{{PSL}}}_{2}\left (\mathbb{F}_{p}\right )$| is |$E/{\mathbb{Q}}(v)$|⁠, and the branch points with ramification index 4 are the roots of the equation u2 − 6u + 25 = 0, where u is related to v via |$v:=\sqrt{p^{\star }}(u+5)/({u-5})$|⁠. From the above data, one computes that the fields |${\mathbb{Q}}(v_{i})$|⁠, where vi is a branch point of ramification index 4, do not contain |$\sqrt{-1}$|⁠. In fact, one has |$v_{i}=\pm 2\sqrt{-p^{\star }}$|⁠. Clearly, all of the above remains true for the extension E ⋅ k/k(v) instead of |$E/{\mathbb{Q}}(v)$|⁠. In particular, as k contains neither |$\sqrt{-1}$| nor |$\sqrt{p^{\star }}$|⁠, one has |$\sqrt{-1} \not \in k(v_{i})$|⁠. By Lemma 2.3, the residue degree at a branch point of E ⋅ k/k(v) with ramification index 4 is divisible by 2. Hence, the decomposition group at such a branch point is of order divisible by 8. It then remains to apply Theorem 5.2 to conclude. 6 Application to the Hilbert-Grunwald Property The present section is devoted to our application to the Hilbert-Grunwald property. We state and prove Theorem 6.2, which is a more general version of Theorem 1.5, and show that it applies to any finite non-abelian simple group; see Corollary 6.5. For this section, let k be a number field, R its ring of integers, T an indeterminate over k, and G a finite group. Given finitely many k-regular G-extensions E1/k(T), |$\dots $|⁠, Es/k(T), we are interested in the following question, for which the partial positive answer provided by Theorem 4.4 is a natural motivation: Question 6.1. Does there exist a finite set |$\mathcal{S}_{\textrm{{exc}}}$| of non zero prime ideals of R such that every Grunwald problem |$(G, (L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$|⁠, with |$\mathcal{S}$| disjoint from |$\mathcal{S}_{\textrm{{exc}}}$|⁠, has a solution inside the union of the sets of specializations of E1/k(T), |$\dots $|⁠, Es/k(T)? It follows from earlier works by the first two authors [22, 24] that, for each nontrivial finite group G for which there exists a k-regular G-extension of k(T), there always is one of these realizations for which the answer to Question 6.1 is negative. In Theorem 6.2 below, we show that the answer to Question 6.1 is in fact always negative for many finite groups: Theorem 6.2. Assume that G has a non cyclic abelian subgroup. Then, the answer to Question 6.1 is negative for all finite sets of k-regular G-extensions of k(T). The proof of Theorem 6.2 requires the following auxiliary result, which strengthens the special case of Theorem 4.6 where τi(j) is chosen to be the identity element of |$D_{t_{i(j)}}$|⁠. Proposition 6.3. Let E/k(T) be a k-regular G-extension, and F the compositum of the residue fields |$\left (E(t_{1})\right )_{t_{1}}, \dots , (E(t_{r}))_{t_{r}}$| of E/k(T) at the branch points |$t_{1},\dots ,t_{r}$|⁠. Moreover, let |${\mathfrak{p}}$| be a prime ideal of R that is totally split in F/k (avoiding a finite set of primes depending only on E/k(T)). Then, the decomposition group of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| is cyclic for every |$t_{0} \in \mathbb{P}^{1}(k)$|⁠. Proof. Let t0 be in |$\mathbb{P}^{1}(k)$|⁠. First, assume that t0 is in |$\{t_{1},\dots ,t_{r}\}$|⁠. Then, by the definition of F, the decomposition group |$D_{t_{0},{\mathfrak{p}}}$| of |$E_{t_{0}}/k$| at |${\mathfrak{p}}$| is trivial. Now, assume that t0 is not in |$\{t_{1},\dots ,t_{r}\}$|⁠. If t0 does not meet any branch point of E/k(T) modulo |${\mathfrak{p}}$|⁠, then, by the Specialization Inertia Theorem (Section 3), |${\mathfrak{p}}$| is unramified in |$E_{t_{0}}/k$| (up to excluding finitely many primes of k). Hence, |$D_{t_{0},{\mathfrak{p}}}$| is cyclic. So we may assume that t0 meets some branch point ti modulo |${\mathfrak{p}}$|⁠. As |${\mathfrak{p}}$| is totally split in F/k, part (1) of Theorem 4.1 shows that |$D_{t_{0},{\mathfrak{p}}}$| is contained in the inertia group |$I_{t_{i}}$| of E(ti)/k(ti)(T) at (T − ti) k(ti)[T − ti] (up to excluding finitely many primes of k). As |$I_{t_{i}}$| is cyclic, we are done. Proof of Theorem 6.2 Given a positive integer s, let |$\left \{E_{1}/k(T), \dots , E_{s}/k(T)\right \}$| be a finite set of k-regular G-extensions. Let F be the compositum of the residue extensions at branch points of |$E_{1}/k(T), \dots , E_{s}/k(T)$|⁠. Then, by Proposition 6.3, for every non zero prime ideal |${\mathfrak{p}}$| of R that is totally split in F/k (outside some finite set |$\mathcal{S}_{{\textrm{{exc}}}}$| depending only on |$E_{1}/k(T),\dots ,E_{s}/k(T)$|⁠), the completion at |${\mathfrak{p}}$| of every specialization of any of the extensions |$E_{1}/k(T), \dots , E_{s}/k(T)$| has cyclic Galois group. Let H be a non cyclic abelian subgroup of G. Without loss of generality, we may assume |$H=\mathbb{Z}/q\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$| for some prime number q. Let ζq be a primitive q-th root of unity, and let |${\mathfrak{p}}$| be a non zero prime ideal of R, not in |$\mathcal{S}_{{\textrm{{exc}}}}$|⁠, that is totally split in |$F\left (\zeta _{q}\right )/k$|⁠. As |${\mathfrak{p}}$| is totally split in |$k\left (\zeta _{q}\right )/k$|⁠, there exists a finite Galois extension |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| with Galois group H. This domain is the completion of R′[S] at the maximal ideal generated by |${\mathfrak{p}}^\prime $| and S; see, for example, [13, Exercise 7.11]. In particular, |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| cannot occur as the completion at |${\mathfrak{p}}$| of a specialization of any of the extensions |$E_{1}/k(T), \dots , E_{s}/k(T)$|⁠, completing the proof. Remark 6.4. Our method cannot provide more examples of finite groups such that the answer to Question 6.1 is negative. Indeed, it requires the following assumption on G: (*) G has a non cyclic subgroup H such that, for every number field F containing k, there exist infinitely many non zero prime ideals |${\mathfrak{p}}$| of R such that |${\mathfrak{p}}$| is totally split in F/k, and such that H occurs as a Galois group over |$k_{\mathfrak{p}}$|⁠. As shown in the proof of Theorem 6.2, condition (*) holds if G has a non cyclic abelian subgroup. However, the converse is true, as it is easy to see that any subgroup H of G as in condition (*) has to be abelian. As an immediate consequence of Theorem 6.2, the answer to Question 6.1 is always negative for finite non cyclic abelian groups, dihedral groups Dn (of order 2n) with n even, symmetric groups Sn with n ≥ 4, alternating groups An with n ≥ 4. In Corollary 6.5 below, we show that the same is true for arbitrary finite non-abelian simple groups: Corollary 6.5. Assume that G is a finite non-abelian simple group. Then, the answer to Question 6.1 is negative for all finite sets of k-regular G-extensions of k(T). Proof. By Theorem 6.2, it suffices to show that G has a non cyclic abelian subgroup. Suppose by contradiction that G does not. Then, by, for example, [6, Chapter XI, Theorem 11.6], every Sylow subgroup of G is either cyclic or a generalized quaternion group. If every 2-Sylow subgroup of G was cyclic, then, every Sylow subgroup of G would then be cyclic. As Hölder proved in 1895 (see, e.g., [31, Theorem 7.53] for more details), G would be solvable, which cannot happen. One may then apply [36, Theorem C] to get that G has a normal subgroup H, which satisfies the following two conditions: (1) (G : H) ≤ 2, (2) |$H = H^{\prime} \times{\textrm{{SL}}}_{2}(\mathbb{F}_{p})$| for some prime number p and some subgroup H′ of H whose Sylow subgroups are cyclic. If (G : H) = 2, then, H is trivial as G is simple. Hence, G has order 2, which cannot happen. Then, by (1), we get |$G = H^{\prime} \times{\textrm{{SL}}}_{2}(\mathbb{F}_{p})$|⁠, with H′ and p as in (2). As G is simple and |${\textrm{{SL}}}_{2}\left (\mathbb{F}_{p}\right ) \not = \{1\}$|⁠, we get H′ = {1}, that is, |$G={\textrm{{SL}}}_{2}(\mathbb{F}_{p})$|⁠. Hence, |${\textrm{{SL}}}_{2}(\mathbb{F}_{p})$| is simple, which cannot happen as |${\textrm{{SL}}}_{2}\left (\mathbb{F}_{2}\right ) \cong S_{3}$|⁠, and, for p ≥ 3, the center of |${\textrm{{SL}}}_{2}(\mathbb{F}_{p})$| has order 2. Solvability of Grunwald problems via specialization is being investigated further by the first author, in a context of infinite families of regular extensions. See [21]. 7 Application to Finite Parametric Sets This section is devoted to our application to the nonexistence of finite parametric sets over number fields, as already mentioned in Section 1.3.3. In Section 7.1, we state and prove Theorem 7.2, which is our new general criterion to provide examples of finite groups with no finite parametric set over number fields. Explicit examples, including those given in Theorem 1.6 from the introduction (Section 1.3.3), are then given in Section 7.2. For this section, let k be a number field, R its ring of integers, T an indeterminate over k, and G a finite group. Let us recall the following definition [18]: Definition 7.1. We say that a set S of k-regular G-extensions of k(T) is parametric if every G-extension of k occurs as a specialization of some extension E/k(T) in S. It is shown in [18] that certain finite groups do not possess finite parametric sets over the given number field k. However, the “global” strategy developed in that paper requires such finite groups to have a non trivial proper normal subgroup which satisfies some further properties. In particular, this cannot be used for finite simple groups. 7.1 A new general criterion for non-parametricity Here, we rather use a “local” approach. Namely, in Theorem 7.2 below, we provide a sufficient condition, based on the proof of Theorem 6.2, for the finite group G to have no finite parametric set over k. Given a prime number q, let ζq denote a primitive q-th root of unity. Theorem 7.2. Suppose that the following condition holds: (**) there exists a prime number q and a number field F containing |$k\left (\zeta _{q}\right )$| such that, for all but finitely non zero prime ideals |${\mathfrak{p}}$| of R which are totally split in F/k, there exists a G-extension of k whose completion at |${\mathfrak{p}}$| has Galois group |$\mathbb{Z}/q\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$|⁠. Then, given a finite set S of k-regular G-extensions of k(T), there exist infinitely many G-extensions of k each of which is a specialization of no extension E/k(T) in S. In particular, G has no finite parametric set over k. Proof. Given a positive integer s, let |$E_{1}/k(T), \dots , E_{s}/k(T)$| be k-regular G-extensions. Pick a prime number q and a number field F as in condition (**). In particular, |$\mathbb{Z}/q\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$| occurs as a subgroup of G. By Proposition 6.3, there exist infinitely many prime ideals |${\mathfrak{p}}$| of R which are totally split in F/k, and such that the Galois group of the completion at |${\mathfrak{p}}$| of any specialization of Ei/k(T) is cyclic (⁠|$i=1,\dots ,s$|⁠). Indeed, one may take all prime ideals |${\mathfrak{p}}$| which are totally split in the compositum of F and all the residue extensions at branch points of Ei/k(T), |$i=1,\dots ,s$| (up to finitely many exceptions depending only on |$E_{1}/k(T), \dots , E_{s}/k(T)$|⁠). In particular, for such a |${\mathfrak{p}}$|⁠, a G-extension of k as in condition (**) is a specialization of Ei/k(T) for no |$i \in \{1,\dots ,s\}$|⁠, completing the proof. 7.2 Explicit examples Corollary 7.3 below contains our main examples of finite groups with no finite parametric set over number fields: Corollary 7.3. Assume that either one of the following two conditions holds: (1) G has a non cyclic abelian subgroup, and there exists a finite set |$\mathcal{S}_{\textrm{{exc}}}$| of non zero prime ideals of R such that every Grunwald problem |$(G,(L^{({\mathfrak{p}})}/k_{\mathfrak{p}})_{{\mathfrak{p}}\in \mathcal{S}})$|⁠, with |$\mathcal{S}$| disjoint from |$\mathcal{S}_{\textrm{{exc}}}$|⁠, has a solution. (2) G = An (n ≥ 4). Then, G has no finite parametric set over k. Proof. In either case, it suffices to show that condition (**) of Theorem 7.2 holds. This is clear in case (1). Now, assume that we are in case (2). Then, |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| occurs as a subgroup of G. It then suffices to show that, for all prime ideals |${\mathfrak{p}}$| of R, there exists a G-extension of k whose completion at |${\mathfrak{p}}$| has Galois group |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠. Let |${\mathfrak{p}}$| be a prime ideal of R, and let |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$| be a |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|-extension. Pick a |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|-extension L/k whose completion at |${\mathfrak{p}}$| is |$L^{({\mathfrak{p}})}/k_{\mathfrak{p}}$|⁠. By a classical result of Mestre (see [26] and [19, Theorem 3]), L/k occurs as a specialization of some k-regular An-extension E/k(T). More precisely, there is a polynomial P(T, Y ) ∈ k[T][Y ] which is monic and separable in Y, with splitting field E over k(T), and such that P(0, Y ) is separable with splitting field L over k. Lemma 2.2 now ensures the existence of t0 ∈ k such that the specialization of E/k(T) at t0 has Galois group An and has the same completion at |${\mathfrak{p}}$| as L/k. This completes the proof. Remark 7.4. (1) Aside from alternating groups, we are not aware of any other infinite family of non-abelian simple groups which have been shown to satisfy condition (**) of Theorem 7.2. However, by using the same tools, the following variant is derived: let G be a finite non-abelian simple group, and let p be a prime number such that |$H:=\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$| occurs as a subgroup of G. Such a prime number p exists by the proof of Corollary 6.5. Then, given a finite set S of k-regular G-extensions of k(T), there exist infinitely many H-extensions of k each of which is a specialization of E/k(T) for no E/k(T) ∈ S. (2) Under the expectation of [16, Section 1] mentioned in Section 1.1, condition (1) of Corollary 7.3 holds (and then the conclusion that G has no finite parametric set over k holds as well), provided G is solvable and has a non cyclic abelian subgroup. 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