Tag - Automorphic forms

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.

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.

Arno Kret: Galois Representations for the general symplectic group

30th March, 2017

In a recent preprint with Sug Woo Shin I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

Yihang Zhu: The Hasse-Weil zeta functions of the intersection cohomology of minimally compactified orthogonal Shimura varieties

20th October, 2016

Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before. In the particular case of orthogonal Shimura varieties, we discuss the computation of the Frobenius-Hecke traces on the intersection cohomology of their minimal compactifications, and the comparison to the Arthur-Selberg trace formula via the process of stabilization. Key ingredients include comparing Harish Chandra character formulas to Kostant's theorem on Lie algebra cohomology, and a comparison between different normalizations of the transfer factors for real endoscopy to get all the signs right.