Talks and Abstracts

On weighted one-level densities of families of L-functions.

Sandro Bettin, University of Genoa

We discuss the one level density of families of L-function, weighted by the central value of the corresponding L-function. We discuss a conjecture of Fazzari for general families with particular attention to the case of the Riemann zeta-function. This is a joint work with Alessandro Fazzari.

Weyl sums with multiplicative coefficients and joint equidistribution

Matteo Bordignon, KTH Stockholm

In this talk I will present a joint work with Cynthia Bortolotto and Bryce Kerr, where we generalize a result of Montgomery and Vaughan regarding exponential sums with multiplicative coefcients to the setting of Weyl sums. As applications, we establish a joint equidistribution result for roots of polynomial congruences and polynomial values and obtain some new results for mixed character sums.

Exact formulas and log-concavity for unimodal sequences

Kathrin Bringmann, University of Cologne

In my talk I describe how modularity can be used to obtain exact formulas and logconcavity results for unimodal sequenes and related parition functions.      

Some new equidistribution results for tori in GSp(4)

Farrell Brumley, Université Sorbonne Paris Nord

Linnik’s theorem on the equidistribution of CM points of large discriminant on the modular curve, subject to a splitting condition, was generalized to the case of anisotropic tori in GL3 by Einsiedler, Lindenstrauss, Michel, and Venkatesh, using a beautiful mix of ergodic and automorphic methods. The number theoretic input in their work comes in the form of subconvex bounds on Dedekind zeta functions for cubic fields. The latter arise from the Hecke formula, which unfolds the toric period of a mirabolic Eisenstein series. One of the obstacles to extending their methods to groups other than GL(n) is the lack of such formulae providing a bridge to L-functions. In work in progress with Jasmin Matz, we have found a substitute for this number theoretic input for the analogous problem for GSp(4), and can derive some preliminary equidistribution results from it, which we shall describe in this talk.

Coprime-universal quadratic forms

Giacomo Cherubini, INdAM and Sapienza University of Rome

Given a prime p > 3, we prove that there are explicit sets Sp of positive integers, whose cardinality does not exceed 31 and whose elements do not exceed 290, such that a positive definite integral quadratic form is coprime-universal with respect to p (i.e., it represents all positive integers coprime to p) if and only if it represents all the elements in Sp. This generalizes works of Bhargava and Hanke (p = 1, i.e., no coprimality conditions), Rouse (p = 2), and De Benedetto and Rouse (p = 3). The proof is based on algebraic and analytic methods, plus a large computational part. Joint work with Matteo Bordignon.

On the distribution of closed geodesic cycle integrals and non-vanishing of central values of L-functions

Petru Constantinescu, École Polytechnique Fédérale de Lausanne

The study of the distribution of Heegner points and closed geodesics is an important and rich subject in analytic number theory, at the interface of automorphic forms, geometry and homogeneous dynamics. In this work, we look at residual distribution of closed geodesics associated to narrow class groups of real quadratic fields. Via Waldspurger’s formula, we obtain applications to non-vanishing of central values of Rankin-Selberg L-functions. Joint work with Asbjørn Nordentoft.

Extending the unconditional support in an Iwaniec–Luo–Sarnak family

Daniel Fiorilli, CNRS Université Paris-Saclay

This is joint work with Lucile Devin and Anders Södergren. In the highly influential work of Iwaniec, Luo and Sarnak, the Katz-Sarnak prediction for the one-level density has been confirmed in several families of holomorphic cusp form L-functions for certain test functions. In the family of newforms of fixed even weight and squarefree level tending to infinity, Iwaniec, Luo and Sarnak proved this prediction unconditionally when the support of the Fourier transform of the implied test function is contained in the interval (−3/2, 3/2), and under GRH in the extended interval (−2, 2). In this talk I will discuss how one can extend the unconditional admissible support to the interval (−1.866..., 1.866...) for a weighted version of the one-level density when the level grows to infinity through prime values. The main new tool in our analysis is the use of zero-density estimates for Dirichlet L-functions.

A supplement to Chebotarev’s density theorem

Gergely Harcos, Alfréd Rényi Institute of Mathematics

In the first part of the talk, I will present a streamlined proof of the Chebotarev density theorem using Frobenius reciprocity and Artin reciprocity. In the second part, I will show how the ideas of the proof lead to related interesting statements, some of which are new. Based on recent joint work with Soundararajan, and the earlier work of Heilbronn (1973) and Foote-Murty (1989).

Restricted Arithmetic Quantum Unique Ergodicity

Peter Humphries, University of Virginia

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface.

Bounds for Averages of Shifted Convolution Sums

Ikuya Kaneko, Caltech

I will discuss averages of shifted convolution sums associated to Hecke–Maaß cusp forms for GL(3) via the circle method, Voronoï and Poisson summation, and a finite field analogue of van der Corput differencing for exponential sums. Immediate consequences of the main result include nontrivial bounds for the second moment of GL(3) L-functions in the t-aspect and the q-aspect as well as bootstrapped subconvex bounds for self-dual GL(3) × GL(2) L-functions in the spectral aspect. If time permits, I will also highlight some arithmetic implications for zero density estimates and the Rankin–Selberg problem. This is joint work with Maksym Radziwiłł at the University of Texas at Austin.

The Weyl law with uniform power savings

Didier Lesesvre, Université de Lille

For a compact Riemannian manifold, the Weyl law describes the asymptotic behavior of the number of eigenvalues of the underlying Laplace operator. Understanding lower order or error terms remains particularly challenging. In the more general context of locally symmetric spaces, the spectral theory of the Laplacian is intimately related to the theory of automorphic forms (among which are elliptic curves, modular or Maass forms, Galois representations...) and similar questions arise.

It is therefore natural to ask for such a Weyl law to hold for families of all automorphic forms of a given reductive group. Until recently, however, all the known asymptotics were for automorphic forms with fixed aspects. In some sense, this amounts to picking a "slice" of the space of automorphic forms only. Unfortunately, making explicit the hidden dependencies in the featured error term does not allow to sum over these aspects to obtain a uniform counting law: existing results did not allow to patch back together the slices.

In their recent achievement, Brumley and Milicevic obtained a uniform Weyl law for GL(2), using the trace formula of Arthur, but with an error term saving only by a power of log. Simplifying the very general setting of this work, and going back to ideas used a long time ago by Drinfeld in the setting of function fields, we obtained a power savings in the smooth Weyl law for the universal family of all automorphic forms of GL(2). The idea is to study a suitable "conductor zeta function", and to deduce a counting law by Tauberian arguments, mimicking a standard strategy in the realm of counting rational points on varieties.

L1 norms of exponential sums with multiplicative coefficients

Mayank Pandey, Princeton University

We show lower bounds for the L1-norm of exponential sums with multiplicative coefficients, fully answering this question up to pretentiousness. We also, with separate methods, lower bound the L1-norm of the exponential sum with the Liouville function, matching unconditionally the lower bound one can obtain on GRH. Joint work with Maksym Radziwill.

Counting and equidistribution

Yiannis Petridis, University College London

I will discuss how counting orbits in hyperbolic spaces lead to interesting number theoretic problems. The counting problems (and the associated equidistribution) can be studied with various methods, and I will emphasize automorphic form techniques, originating in the work of H. Huber and studied extensively by A. Good. My collaborators in various aspects of this project are Chatzakos, Lekkas, Nordentoft, Risager, and Voskou.

On a mean value result for a product of L-functions

Nicole Raulf, Université de Lille

The asymptotic behaviour of moments of L-functions is of special interest to number theorists and there are conjectures that predict the shape of the moments for families of L-functions of a given symmetry type. However, only some results for the first few moments are known. In this talk we will consider the asymptotic behaviour of the first moment of the product of a Hecke L-function and a symmetric square L-function in the weight aspect. This is joint work with O. Balkanova, G. Bhowmik, D. Frolenkov.

Prime length counting and equidistribution in hyperbolic groups

Morten Risager, University of Copenhagen

We investigate various counting and equidistribution problems in hyperbolic groups, focusing in particular at what happens when we are only counting according to prime lengths. This is work in progress with Yiannis Petridis. 

An estimate of Linnik revisited

Stelios Sachpazis, Université de Montréal

Let a and q be two coprime positive integers. In 1944, Linnik proved his celebrated theorem concerning the size of the smallest prime p(q, a) in the arithmetic progression a(mod q). In his attempt to prove this result, Linnik established an estimate for the sums of the von Mangoldt function Λ on arithmetic progressions. His work on p(q, a) was later simplified, but the simplified proofs relied in one form or another on the same advanced tools that he originally used. The last two decades, some more elementary approaches for Linnik’s theorem have appeared. Nonetheless, none of them furnishes an estimate of the same quantitative strength as the ones that the classical methods obtain for the sums of Λ on arithmetic progressions. In this talk, we will see how one can seal this gap and recover an analogue of Linnik’s estimate, that is a strongly uniform estimate for primes in arithmetic progressions, by largely elementary means. The ideas that I will describe build on methods from the treatment of Koukoulopoulos on multiplicative functions with small partial sums and his pretentious proof of the prime number theorem in arithmetic progressions.

Global sup-norm bounds for Siegel cusp forms of degree 2

Abhishek Saha, Queen Mary University of London

We prove a global sup-norm bound in the weight aspect for Siegel cusp forms of degree 2 and full level, assuming the GRH. The method relies on the Fourier expansion, and as a key step towards our result we bound the Fourier coefficients in terms of certain central L-values. This in turn requires us to bound certain local factors occurring in the refined Gan–Gross–Prasad formula, which is the technical heart of our proof. This is joint work with Félicien Comtat and Jolanta Marzec-Ballesteros.

The second moment method for rational points

Efthymios Sofos, University of Glasgow

In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel’s Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen [arXiv:2212.10373] that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.

Fourth moments of automorphic forms and an application to diameters of hyperbolic surfaces

Raphael S. Steiner, ETH Zürich

In joint work with Ilya Khayutin and Paul Nelson, we demonstrate how theta functions may be used to derive geometric expressions for fourth moments of automorphic forms on hyperbolic surfaces. By carefully estimating a second moment matrix count, we obtain a sharp pointwise bound on the fourth moment in the weight and level aspect. As a consequence, we significantly improve the sup-norm bounds in these aspects and give an unconditional upper bound on the diameter of hyperbolic surfaces of the same strength as if one were to assume the Selberg eigenvalue conjecture.

Hilbert modular groups and Hilbert-Maass forms

Fredrik Strömberg, University of Nottingham

We present some theoretical and computational aspects of Hilbert-Maass forms, that is non-holomorphic modular forms on Hilbert modular groups. In particular we will present a recent reduction algorithm for general Hilbert modular groups and how it can be used to extend the ‘automorphy’ (aka Hejhal’s) algorithm to the setting of Hilbert modular groups.

An explicit Chebotarev Density Theorem on average and applications

Ilaria Viglino, ETH Zürich

The study of specific families of polynomials and their splitting fields can provide useful examples and evidences for conjectures regarding invariants related to number field extensions and related objects. The main example that can illustrate the relevance of the work, is the family P_{n,N}^0 of degree n monic polynomials f with integer coefficients, so that the maximum of the absolute values of the coefficients is less or equal than N , and the splitting field Kf over the rationals Q is the full symmetric group S_n. We let N → +∞. With a little work, this can actually be generalized to polynomials with integral coefficients in a fixed number field of degree d over Q. Let π_{f,r}(x) be the function counting the primes less or equal than x such that f in P_{n,N}^0 has a fixed square-free splitting type r modulo p. It turns out that the quantity (πf,r(x) − δ(r)π(x)((δ(r) − δ(r)2)π(x))−1/2 is distributed like a normal distribution with mean 0 and variance 1, whenever x is small compared to N , e.g. x = N^{1/ log log N}. Here δ(r) is the coefficient in the asymptotic predicted by the classical Chebotarev theorem. This result leads to interesting applications, as finding upper bounds for the torsion part of the class number in terms of the absolute discriminant, as it was done for other infinite families by Ellenberg, Venkatesh and Heath-Brown, Pierce.

The arithmetic volume of moduli space of Abelian surfaces

Anna-Maria Von Pippich, University of Konstanz

Let A_g denote the moduli stack of principally polarized abelian varieties of dimension g. The arithmetic volume of \overline{A_g} is defined to be the arithmetic degree of the metrized Hodge bundle on \overline{A_g}. In 1999, Kühn proved a beautiful formula for the arithmetic height of \overline{A_1} in terms of special values of the Riemann zeta function. In this talk, we report on joint work with Barbara Jung generalizing Kühn’s result to the case g = 2.

Around the Gauss circle problem: Hardy’s conjecture and the distribution of lattice points near circles

Igor Wigman, King’s College London

This talk is based on a joint work with Steve Lester (KCL). Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-R disc by its area is O (R^{1/2+o(1)}). One source of support for this conjecture is a folklore heuristic that uses i.i.d. random variables to model the lattice points lying near the boundary and square-root cancellation of sums of these random variables. We examine this heuristic by studying how these lattice points interact with one another and prove that their autocorrelation is determined in terms of a random model. Additionally, it is shown that lattice points near the boundary which are "well separated" behave independently. We also formulate a conjecture concerning the distribution of pairs of these lattice points.