Tag - Analytic number theory

John Bergdall: Upper bounds for constant slope p-adic families of modular forms

28th January, 2019

This talk is concerned with the radius of convergence of p-adic families of modular forms: q-series over a p-adic disc whose specialization to certain integer points is the q-expansion of a classical Hecke eigenform of level p. Numerical experiments by Gouvêa and Mazur in the 1990s predicted the general existence of such families but also suggested, in spirit, the radius of convergence in terms of an initial member. Buzzard and Calegari showed, ten years later, that the Gouvêa-Mazur prediction was false. It has since remained open question how to salvage it. Here we will present some recent theoretical results towards such a salvage, backed up by numerical data.

Lynnelle Ye: Slopes in eigenvarieties for definite unitary groups

6th December, 2018

The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing p-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the Up Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension 2, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions. We will then discuss the ideas of the proof, which goes through the classification of automorphic representations that are principal series at p, and a geometric consequence.

Will Sawin: The Lucky Logarithmic Derivative

29th November, 2018

We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials. This method is based on work of Hast and Matei, except that thanks to a trick involving the logarithmic derivative, we are able to achieve a much smaller error term than is possible by this method for a "typical" problem of this type.

Samit Dasgupta: Explicit formulae for Stark Units and Hilbert’s 12th problem

11th October, 2018

Hilbert's 12th problem is to provide explicit analytic formulae for elements generating the maximal abelian extension of a given number field. In this talk I will describe an approach to Hilbert's 12th that involves proving exact p-adic formulae for Gross-Stark units. This builds on prior joint work with Kakde and Ventullo in which we proved Gross’s conjectural leading term formula for Deligne-Ribet p-adic L-functions at s=0. This is joint work with Mahesh Kakde.

Jack Buttcane: Non-spherical Poincaré series, cusp forms and L-functions for GL3

10th April, 2018

The analytic theory of Poincaré series and Maass cusp forms and their L-functions for SL3(ℤ) has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for L2(SL3(ℤ)\PSL3(ℝ)/SO3(ℝ)). I will describe the Maass cusp forms of L2(SL3(ℤ)\PSL3(ℝ)) which are minimal with respect to the action of the Lie algebra and give a (relatively) simple method for constructing Kuznetsov-type trace formulas by considering Fourier coefficients of certain Poincaré series. In recent work with Valentin Blomer, we have extended our proof of spectral-aspect subconvexity for L-functions of SL3(ℤ) Maass forms to the non-spherical case, and I will discuss the structure of that proof, as well.

Jayce Getz: Summation formulae and speculations on period integrals attached to triples of automorphic representations

27th March, 2018

Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality.  Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces (Vi,Qi) of even dimension by the equation Q1(v1)=Q2(v2)=Q3(v3). I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of GLn1 × GLn2 × GLn3 where the ni are arbitrary, and speculate on the relationship between these period integrals and Langlands L-functions.

Ian Petrow: The Weyl law for algebraic tori

13th March, 2018

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

Jun Su: Automorphy for coherent cohomology of Shimura varieties

5th December, 2017

We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic representations. Consequently, any Galois representation attached to these coherent cohomology should be automorphic. Our proof is based on Franke’s work on singular cohomology of locally symmteric spaces and via Faltings' BGG spectral sequence we’ve also strengthened Franke’s result in the Shimura variety case.