Tag - Analytic number theory

Vesselin Dimitrov: Modular forms and arithmetic algebraization methods I

8th April, 2023

We survey some recent developments in the theory of vector-valued modular forms for SL2(ℤ), focusing especially on our recent and ongoing joint work with Frank Calegari and Yunqing Tang that proved the Unbounded Denominators conjecture as one application.

The first talk will be an introduction to noncongruence modular forms, from one side, and from another side to arithmetic algebraization methods. We will discuss how to connect these two subjects, and the kind of further applications that arithmetic algebraization methods may have to offer in number theory. After the basic examples and some history, we will turn to Bost's slopes method of Arakelov theory for the technical underpinning of our proofs.

In the second talk, I will establish a new equivariant holonomy bound and apply it to prove the Unbounded Denominators conjecture of Atkin, Swinnerton-Dyer, and Mason. This will be a new argument alternative to our original proof in (F. Calegari, V. Dimitrov, Y. Tang: The unbounded denominators conjecture).

Paul Bourgade: Random matrices, the Riemann zeta function and branching processes II

8th April, 2023

Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.

Paul Bourgade: Random matrices, the Riemann zeta function and branching processes I

8th April, 2023

Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.

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.

Alexandre de Faveri: Simple Zeros of Modular Form L-Functions

3rd April, 2023

Zeros of L-functions have been extensively studied, due to their close connection to arithmetic problems. Despite several precise conjectures about their behaviour, our unconditional understanding of them remains limited. In this talk we will discuss certain intrinsic properties of such zeros, focusing on what is known (in degrees 1 and 2) about their accumulation on the central line and their multiplicity. Here the tools of analytic number theory can give quantitative advances, and we will show how to deduce that there are many zeros of multiplicity one for the L-function associated to a modular form.

Alex Wilkie: Integer points on analytic sets

9th March, 2023

In 2004 I proved an O(log log H) bound for the number of integer points of height at most H lying on a globally subanalytic curve. (The paper was published in the Journal of Symbolic Logic and so probably escaped the notice of most of you reading this.) Recently, Gareth Jones and Gal Binyamini proposed a generalization of the result to higher dimensions (where the obvious statement is almost certainly false) and I shall report on our joint work: one obtains the (hoped for) (log log H)n bound for (not globally subanalytic but) globally analytic sets of dimension n.

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.

Chris Skinner: An Euler System for the Symmetric Square of a Modular Form

16th February, 2023

I will explain a new construction of an Euler system for the symmetric square of an eigenform and its connection with L-values. The construction makes use of some simple Eisenstein cohomology classes for Sp(4) or, equivalently, SO(3,2). This is an example of a larger class of similarly constructed Euler systems. This is a report on joint work with Marco Sangiovanni Vincentelli.

Xiannan Li: Quadratic twists of modular L-functions

8th December, 2022

The behaviour of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann hypothesis by the work of Soundararajan and Young. Our proof depends on deriving an optimal large sieve type bound.

Kevin Ford: Toward a Theory of Prime Detecting Sieves

17th November, 2022

Given a set of integers, we wish to know how many primes there are in the set. Modern tools allow us to obtain an asymptotic for the number of primes, or at least a lower bound of the expected order, assuming certain strength Type-I information (the distribution of the sequence in progressions) and Type-II information (bilinear sums over the sequence). The methods used previously, especially Harman's sieve, are largely ad-hoc and shed little light on the limitations of the methods. In joint work with James Maynard, we develop a systematic framework for understanding the theoretical limits of these prime detecting sieves, which allow us, in principle, to answer these questions for any given Type I and Type-II information.