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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.

Youness Lamzouri: A walk on Legendre paths

15th June, 2023

In this talk, we shall explore certain polygonal paths, that we call ''Legendre paths'', which encode important information about the values of the Legendre symbol. More precisely, the Legendre path modulo a prime number p is defined as the polygonal path in the plane whose vertices are the points (j, Sp(j)) for 0≤j≤p-1, where Sp(j) is the (normalized) sum of Legendre symbols (n/p) for n up to j. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? when does a Legendre path decreases for the first time? what is the typical number of x-intercepts of such paths? and what proportion of a Legendre path is above the real axis? We will see that some of these questions correspond to important and longstanding problems in analytic number theory, including understanding the size of the least quadratic non-residue, and improving the Pólya-Vinogradov inequality for character sums. Among our results, we prove that as we average over the primes, the Legendre paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademacher random multiplicative functions. Part of this work is joint with Ayesha Hussain and with Oleksiy Klurman and Marc Munsch.

Carlo Pagano: On Chowla’s non-vanishing conjecture over function fields

8th June, 2023

A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz-Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui-Florea, David-Florea-Lalin, Ellenberg-Li-Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.

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.

Nina Zubrilina: Root Number Correlation Bias of Fourier Coefficients of Modular Forms

20th April, 2023

In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number.

In my talk, I will discuss this root number correlation bias when the average is taken over weight 2 modular newforms of all Galois orbit sizes simultaneously. I will point to a source of this phenomenon in this case and compute the correlation function exactly.

Oleksiy Klurman: On Elliott’s Conjecture and Applications

11th April, 2023

Given any bounded multiplicative function f :ℕ→𝔻, a deep conjecture of Elliott predicts cancellations of the form

limx→∞ (1/x) ∑n≤ x f(n+h1) f(n+h2) ... f(n+hk)=0,

for all distinct shifts hi∈ℕ unless it is 'close' to the modulated Dirichlet character in an appropriate sense. Partial progress towards this conjecture has had numerous consequences, including solution of the Erdős discrepancy problem, progress on the (logarithmic) Chowla and Sarnak's conjectures, and many others. In this talk I will report on some new developments towards this conjecture and present several applications in number theory, ergodic theory and combinatorics. This is based on a joint work with A. Mangerel and J.Teräväinen.

Vesselin Dimitrov: Modular forms and arithmetic algebraization methods II

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).

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.