Tag - Automorphic forms

Jessica Fintzen: An introduction to representations of p-adic groups

25th March, 2024

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory, but also for the construction of an explicit and a categorical local Langlands correspondence, and has applications to the study of automorphic forms, for example. In my talk I will introduce p-adic groups and explain that the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Tony Feng: Mirror Symmetry and the Breuil-Mezard Conjecture

4th December, 2023

The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry.

David Farmer: The landscape of L-functions

10th July, 2023

L-functions of degree d can be parametrized, in two different ways, by points with an attached multiplicity in (d-1)-dimensional Euclidean space. One approach separates the L-functions according to the shape of the Gamma-factors in the functional equation, equivalently, according to the infinity type of the underlying automorphic representation. The other approach combines all the L-functions of a given degree into a single picture in which the points, to leading order, are uniformly dense. We will describe these classifications and provide examples of several 'landscapes' in the L-function world.

Jack Thorne: Symmetric Power Functoriality For Hilbert Modular Forms

18th May, 2023

Symmetric power functoriality is one of the basic cases of Langlands's functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over ℚ, as the prime p varies). I will discuss the proof of the existence of the symmetric power liftings of Hilbert modular forms of regular weight. The proof uses automorphy lifting theorems, automorphic forms on unitary groups, and the geometry of Shimura varieties, as well as the fact that Spec(ℤ) is simply connected.

Simon Marshall: Subconvexity for L-functions on U(n) x U(n+1)

6th April, 2023

We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral. There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for the L-value if the test vector is chosen correctly. We are able to improve the bound for the period integral using a technique called arithmetic amplification, which uses the action of the Hecke operators, and this yields a subconvex bound.

Valentin Blomer: Applications of the Relative Trace Formula

23rd February, 2023

I discuss the spectral and arithmetic side of the relative trace formula of Kuznetsov type for congruence subgroups of SL(n,ℤ) with applications to automorphic density theorems. A particular focus is on properties of general Kloosterman sums as well as the average size of Fourier coefficients in absence of newform theory.

Rahul Dalal: Applications of the Endoscopic Classification to Statistics of Cohomological Automorphic Representations on Unitary Groups

3rd November, 2022

Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation π0 at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to infinity. For unramified unitary groups and a large class of π0, we are able to compute the exact leading term for both counts of representations and averages of Satake parameters. We get bounds on our error term similar to previous work by Shin-Templier that studied the case of discrete series at infinity. We also discuss in-progress computations towards expected corollaries related to the Sarnak-Xue density conjecture, average Sato-Tate equidistribution in families, and growth of cohomology for towers of locally symmetric spaces.

The main technical tool is an extension of an inductive argument that was originally developed by Taïbi to count unramified representations on Sp and SO and used the endoscopic classification of representations (which our case requires for non-quasisplit unitary groups).

This is joint work with Mathilde Gerbelli-Gauthier.

Ngô Bảo Châu: On the kernel of the non-abelian Fourier transform

20th October, 2022

Tate reformulated the theory of the Riemann zeta function and its functional equation as the Mellin shadow of the Fourier transform on a certain space of function on the adeles. Conjecturally, Langlands' general automorphic L-functions and their functional equation can be interpreted in the same way following a framework due to Braverman and Kazhdan with the case of standard L-function associated with automorphic representations of GLn and the standard representation of the dual GLn being well known and due to Godement and Jacquet. This talk is based on a work in progress jointly with Zhilin Luo in which we propose an explicit conjectural construction for the kernel of the non abelian Fourier transform for G=GLn and arbitrary representation of the dual GLn.

Ping Xi: Analytic approaches towards Katz’s problems on Kloosterman sums

8th September, 2022

Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from ℓ-adic cohomology.