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Shai Evra: Optimal strong approximation and the Sarnak-Xue density hypothesis

3rd November, 2022

It is a classical result that the modulo map from SL2(ℤ) to SL2(ℤ/qℤ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL2(ℤ/qℤ) has a lift in SL2(ℤ) with coefficients of size at most qe? A simple pigeonhole principle shows that e is strictly greater than 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL2(ℤ). The proof relies on a density theorem of the Ramanujan conjecture for SL2(ℤ). In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands programme.

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.

Jeffrey C. Lagarias: The Alternative Hypothesis and Point Processes

6th October, 2022

The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that nearly all normalized zero spacings be near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to the long-standing problem of proving there are no exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing Alternative Hypothesis type statistics, which is consistent with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by Tao with slightly different methods.) The talk reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.

Vesselin Dimitrov: Arithmetic holonomy bounds and Apery limits

22nd September, 2022

A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from the theory of non-congruence and vector-valued modular forms. In this talk, I will report on our sequel joint work-in-progress where we extend the scope of these arithmetic holonomy bounds to beneath the framework of finite-index subgroups of SL2(ℤ) and onto the arithmetic theory of certain periods appearing as Apery limits for local systems on the triply punctured projective line. Applications include irrationality proofs, with quantitative bad approximability measures, of the 2-adic realization of ζ(5), the archimedean period L(2,χ−3)−π(log3)/(3√3), and the products of two logarithms log(1−1/m)log(1−1/n) for arbitrary integer pairs n,m with 0<|1−m/n|<ϵ0, where ϵ0 is some positive absolute constant. As a by-product, we find an arithmetic characterization of the logarithm function.

Alexandra Florea: Negative moments of the Riemann zeta function

22nd September, 2022

I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.

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.

Dipendra Prasad: Branching laws: homological aspects

19th May, 2022

This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the Bernstein decomposition.

Spencer Leslie: Modular forms of half-integral weight on exceptional groups

28th April, 2022

Half-integral weight modular forms are classical objects with many important arithmetic applications. In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL2. In this talk, I will outline a theory of modular forms of half-integral weight on double covers of exceptional groups, generalizing the integral weight theory developed by Gross-Wallach, Gan-Gross-Savin, and Pollack. Furthermore, I discuss a particular example of a weight 1/2 modular form on G2 whose Fourier coefficients encode the 2-torsion in the narrow class groups of totally real cubic fields. This is built by studying a remarkable automorphic representation of the double cover of the exceptional group F4.

Liyang Yang: Non-vanishing of twists of GL4 L-functions

14th April, 2022

I will discuss recent work with Maksym Radziwill in which we show that for any fixed tempered cuspidal representation π of GL4 over the rationals, there exist infinitely many primitive characters χ such that the twisted L-function L(s,π×χ) is non-vanishing at the central point s=1/2. I will focus on the proof, which involves a mix of ideas.

Linus Hamann: Some Progress on Categorical Local Langlands

30th March, 2022

Recently, Fargues and Scholze attached a semi-simple L-parameter to any smooth irreducible representation of a p-adic reductive group, realizing the local Langlands correspondence as a geometric Langlands correspondence over the Fargues-Fontaine curve. They conjectured that there should exist an analogue of the geometric Langlands conjecture in this setting, known as the categorical local Langlands correspondence. Concretely, this conjecture translates to the belief that certain Shtuka spaces, generalizing the Lubin-Tate and Drinfeld towers appearing in the work of Harris-Taylor, should have cohomology dictated by the semi-simple L-parameter that they construct. In this talk, we will explain how one can make some progress on this conjecture by showing the Fargues-Scholze correspondence is compatible with known instances of the local Langlands correspondence through global methods, and then using this compatibility together with techniques from geometric Langlands to fully describe the cohomology of these Shtuka spaces in certain cases.