International Mathematics Research Notices

Nadimpalli, Santosh; Dhar, Sabyasachi

doi: 10.1093/imrn/rnae183pmid: N/A

Let $p$ and $l$ be two distinct odd primes, and let $n\geq 2$ be a positive integer. Let $E$ be a finite Galois extension of degree $l$ of a $p$-adic field $F$. Let $q$ be the cardinality of the residue field of $F$. Let $\pi _{F}$ be an integral $l$-adic generic representation of $\textrm{GL}_{n}(F)$, and let $\pi _{E}$ be the base change of $\pi _{F}$. Let $J_{l}(\pi _{F})$ (resp. $J_{l}(\pi _{E})$) be the unique generic component of the mod-$l$ reduction $r_{l}(\pi _{F})$ (resp. $r_{l}(\pi _{E})$). Assuming that $l$ does not divide $|\textrm{GL}_{n-1}(\mathbb{F}_{q})|$, we prove that the Frobenius twist of $J_{l}(\pi _{F})$ is the unique generic subquotient of the Tate cohomology group $\widehat{H}^{0}(\textrm{Gal}(E/F), J_{l}(\pi _{E}))$—considered as a representation of $\textrm{GL}_{n}(F)$.

Sankaran, Siddarth

doi: 10.1093/imrn/rnae179pmid: N/A

We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of “special” divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.

Ciraolo, Giulio; Li, Xiaoliang

doi: 10.1093/imrn/rnae181pmid: N/A

Given $N\geq 2$, we completely classify solutions to the anisotropic $N$-Liouville equation $$ \begin{align*} &-\Delta_N^H\,u=e^u \quad\textrm{in}\ \mathbb{R}^N,\end{align*} $$under the finite mass condition $\int _{\mathbb{R}^{N}} e^{u}\,dx<+\infty $. Here $\Delta _{N}^{H}$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence in the planar case $N=2$, we give an affirmative answer to a conjecture made in [53].

Laga, Jef; Swaminathan, Ashvin A

doi: 10.1093/imrn/rnae184pmid: N/A

Let $\mathcal{F}_{g}$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over ${\mathbb{Q}}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_{g}$ have no rational points except the point at infinity. In this note, we prove the analogue for quadratic points: for each $g\geq 4$, a positive proportion of curves in $\mathcal{F}_{g}$ have no points defined over quadratic extensions except those that arise by pulling back rational points from $\mathbb{P}^{1}$.

doi: 10.1093/imrn/rnae195pmid: N/A

Levi, Matteo; Marzo, Jordi; Ortega-Cerdà, Joaquim

doi: 10.1093/imrn/rnae182pmid: N/A

We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.

Mundinger, Joshua

doi: 10.1093/imrn/rnae186pmid: N/A

We classify twistings of Grothendieck’s differential operators on a smooth variety $X$ in prime characteristic $p$. We prove that isomorphism classes of twistings are in bijection with $H^{2}(X,\mathbb{Z}_{p}(1))$, the degree 2, weight 1 syntomic cohomology of $X$. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.

Franco, Emilio; Hanson, Robert

doi: 10.1093/imrn/rnae187pmid: N/A

Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

Presnova, Ekaterina; Smirnov, Evgeny

doi: 10.1093/imrn/rnae188pmid: N/A

We give a new combinatorial description for Grassmannian Grothendieck polynomials in terms of subdivisions of Gelfand–Zetlin polytopes. Moreover, these subdivisions also provide a description of Lascoux polynomials. This generalizes a similar result on key polynomials by Kiritchenko, Smirnov, and Timorin.

Maynard, James; Pratt, Kyle

doi: 10.1093/imrn/rnae191pmid: N/A

We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few “half-isolated” zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption.