International Mathematics Research Notices

doi: 10.1093/imrn/rnq199pmid: N/A

doi: 10.1093/imrn/rnq200pmid: N/A

doi: 10.1093/imrn/rnq198pmid: N/A

Zakharov, Dmitry

doi: 10.1093/imrn/rnq014pmid: N/A

We construct a discrete analogue of the integrable two-dimensional Dirac operator and describe the spectral properties of its eigenfunctions. We construct an integrable discrete analogue of the modified NovikovVeselov hierarchy. We derive the first two equations of the hierarchy and give explicit formulas for the eigenfunctions in terms of the theta functions of the associated spectral curve.

Xiong, Maosheng

doi: 10.1093/imrn/rnq015pmid: N/A

Let q be a finite field of cardinality q and l 2 be a prime number such that q 1 (mod l). Extending the work of Faifman and Rudnick [6] on hyperelliptic curves, we study the distribution of zeros of zeta functions of curves over q varying over the moduli spaces of cyclic l-fold covers of 1(q) in the limit of large genus. The zeros all lie on a circle, according to the Riemann Hypothesis for curves, and their angles are uniformly distributed. Moreover, the number of angles inside a fixed symmetric interval I is asymptotically a sum of (l 1)/2 identical independent Gaussian random variables, each of which comes naturally from a Dirichlet character and has mean 4gI/(l 1) and variance (4/2)log(2gI). These results continue to hold for shrinking intervals as long as the expected number of angles 2gItends to infinity.

Mimura, Masato

doi: 10.1093/imrn/rnq011pmid: N/A

We prove that for any euclidean ring R and n 6, = SLn(R) has no unbounded quasi-homomorphisms. By Bavards duality theorem, this means that the stable commutator length vanishes on . The result is particularly interesting for R = F[x] for a certain field F (such as ), because in this case the commutator length on is known to be unbounded. This answers a question of M. Abrt and N. Monod for n 6.

Kolř, Martin

doi: 10.1093/imrn/rnq013pmid: N/A

We introduce an alternative approach to a fundamental CR invariantthe Catlin multitype. It is applied to a general smooth hypersurface in n1, not necessarily pseudoconvex. Using this approach, we prove biholomorphic equivalence of models and give an explicit description of biholomorphisms between different models. A constructive finite algorithm for computing the multitype is described. The results can be viewed as providing a necessary step in understanding local biholomorphic equivalence of Levi degenerate hypersurfaces in n1.

Kowalski, Emmanuel; Nikeghbali, Ashkan

doi: 10.1093/imrn/rnq019pmid: N/A

Building on earlier work introducing the notion of mod-Gaussian convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of mod-Poisson convergence. We show in particular how it occurs naturally in analytic number theory in the classical Erds Kac Theorem. In fact, this case reveals deep connections and analogies with conjectures concerning the distribution of L functions on the critical line, which belong to the mod-Gaussian framework, and with analogues over finite fields, where it can be seen as a zero-dimensional version of the KatzSarnak philosophy in the large conductor limit.

Hill, Richard

doi: 10.1093/imrn/rnq042pmid: N/A

Given a linear algebraic group G defined over a number field, Emerton [10] has defined a sequence of p-adic Banach Spaces . These spaces are representations of the group of adele points of G, and may be regarded as spaces of p-adic automorphic forms. The purpose of the current paper is to introduce some new methods for studying the spaces . We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emertons spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems.

Rigot, Sverine; Wenger, Stefan

doi: 10.1093/imrn/rnq023pmid: N/A

We prove non-extendability results for Lipschitz maps with target space being jet spaces equipped with a left-invariant Riemannian distance, as well as jet spaces equipped with a left-invariant sub-Riemannian CarnotCarathodory distance. The jet spaces give a model for a certain class of Carnot groups, including in particular all Heisenberg groups.