### EXPOSÉS

#### Shabnam Akhtari (Pennsylvania State University)

A Quantitative Primitive Element Theorem

## Résumé

Let K be an algebraic number field. The Primitive Element Theorem implies that the number field K can be generated over the field of rational numbers by a single element of K. We call such an element a generator of K. A simple and natural question is « what is the smallest generator of a given number field? » (and how to find it!) In order to express this question more precisely, we will introduce some height functions. Then we will discuss some open problems and some recent progress in this area.

#### Lior Bary-Soroker (Tel-Aviv University)

## Résumé

Hilbert’s irreducibility theorem is a central theorem in arithmetic geometry with numerous applications. A simple quantitative version of the theorem says that if f(x,y) is an irreducible polynomial in 2 variables with rational coefficients, then for almost all integers a, the univariate polynomial f(a,y) is irreducible. In recent years there has been a surge of studies on variants of Hilbert’s irreducibility theorem over algebraic groups; that is, replacing x by the coordinates of an algebraic group G. The talk aims to explore these variants, highlighting their connections to analytic number theory.

#### Sandro Bettin (Università degli studi di Genova)

## Résumé

Using methods from dynamical systems, we determine the asymptotic distribution of modular symbols associated to modular forms of arbitrary weight and level. This includes in particular the case of Eisentein series of arbitrary level which was missing from earlier works on modular symbols of Petridis and Risager, Nordentoft, Lee and Sun, and Bettin and Drappeau. As an application we also determine the distribution of Fourier series with modular coefficients. This is joint work with Sary Drappeau and Jungwon Lee.

#### Alina Bucur (University of California, San Diego)

## Résumé

A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. This is joint work with Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, Ekin Ozman, and Melanie Matchett Wood.

#### Martin Čech (University of Turku)

## Résumé

There are two common approaches to tackle moments of L-functions: either approximations by Dirichlet polynomials, or using multiple Dirichlet series. The second approach is usually based on modifying the multiple Dirichlet series into a « perfect » object, which satisfies certain functional equations. During this talk, we will relate the two approaches and introduce a way to work with the multiple Dirichlet series without modifying it, which leads to a simpler and more natural computations of the known moments. More specifically, we prove an explicit region of meromorphic continuation of the multiple Dirichlet series associated with moments of real Dirichlet L-functions, which will be wide enough to give the first 3 moments with power-saving error term.

#### Vorrapan Chandee (Kansas State University)

## Résumé

We study a large orthogonal family of L-functions associated with holomorphic Hecke newforms of level q, averaged over q \asymp Q. Assuming GRH, we prove a one level density result for this family with the support of the Fourier transform of the test function being extended to be inside (−4,4). This is the joint work with Siegfred Baluyot and Xiannan Li.

#### John Brian Conrey (American Institute of Mathematics)

## Résumé

I will discuss some of my favorite open problems at the frontier of Random Matrix Theory and Analytic Number Theory.

#### Christophe Delaunay (Université de Franche-Comté)

## Résumé

Let $E$ be an elliptic curve over ${\mathbb Z}(t)$. Then for almost all $t_0 \in {\mathbb Z}$ the specialization of $E$ at $t_0$ is an elliptic curve over ${\mathbb Q}$ giving raise of a family of elliptic curves. From Helfgott’s work, the average root number of such a family should be zero except in some very specific condition. The aim of the talk is to focus on the cases where the average root number is not zero. This study is a joint work with Chantal David and Sandro Bettin.

#### Lucile Devin (Université du Littoral Côte d’Opale)

## Résumé

We study the low-lying zeros of a family of L-functions related to a CM elliptic curve, and find out that the family breaks down into two subfamilies of different symmetry types: orthogonal and symplectic.

#### Alexander Dunn (Georgia Institute of Technology)

## Résumé

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki’s evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes. This is a joint work with C.David, A.Hamieh, and H.Lin.

#### Brooke Feigon (The City College of New York)

## Résumé

Expander graphs are sparse, highly connected graphs. They have applications to Computer Science, coding theory, neural networks and other areas. Ramanujan graphs are optimal expander graphs. Lubotzky, Phillips and Sarnak and in separate work Margulis, first constructed infinite families of regular Ramanujan graphs of fixed degree in the late 1980s using tools from number theory. More specifically, these graphs can be viewed as quotients of Bruhat-Tits trees by congruence subgroups of arithmetic lattices in GL(2). They made these graphs explicit by viewing them as Cayley graphs and the Ramanujan property of the graphs follows from the Generalized Ramanujan Conjecture on GL(2). In this talk we will discuss Ramanujan bigraphs, which arise as quotients of the Bruhat-Tits buildings of U(3), where the associated automorphic spectrum is tempered. We will describe the notion of Cayley bigraphs, which we develop in order to describe explicit constructions of Ramanujan bigraphs. Next we will give criteria for automorphic representations to be Ramanujan, give examples of both Ramanujan and non-Ramanujan Cayley bigraphs and if time allows outline some parts of the proofs. This talk is based on joint work with Shai Evra, Kathrin Maurischat and Ori Parzanchevski. Part of this work originated from a project from Women in Numbers Europe 1.

#### Alexandra Florea (University of California Irvine)

## Résumé

Fixing two Dirichlet characters \chi_1 and \chi_2 modulo q (where q is a prime), Zacharias showed that there exists a positive proportion of characters \chi modulo q such that L(1/2, \chi), L(1/2, \chi \chi_1), L(1/2, \chi \chi_2) are simultaneously nonzero. I will talk about recent joint work with H. Bui which extends this result. Namely, I will explain how one can fix three characters \chi_1, \chi_2 and \chi_3 modulo q and obtain the simultaneous nonvanishing of L(1/2,\chi), L(1/2, \chi \chi_1), L(1/2, \chi \chi_2), L(1/2, \chi \chi_3) for a positive proportion of characters \chi modulo q.

#### Andrew Granville (Université de Montréal)

## Résumé

In this talk I will review some of Chantal’s work taking an analytic perspective on questions arising from algebraic number theory.

#### Ahmet Güloğlu (Bilkent University)

TBC

## Résumé

TBC

#### Alia Hamieh (University of Northern British Columbia)

Large Sums of Fourier Coefficients of Cusp Forms

## Résumé

Consider the partial sum $S(x,f)=\sum_{n\leq x}\lambda_f(n)$ associated with the Fourier coefficients of a primitive form $f$ of weight $k$ and level $N$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq (kN)^{\epsilon}$. Lamzouri proved that this is true under the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this talk, we show that this conjecture holds under a weaker assumption than GRH. We also discuss some implications of $S(x,f)$ being large. Our work extends results of Granville and Soundararajan on large character sums. This work originated from the Women In Numbers 6 Workshop and is joint with Claire Frechette, Mathilde Gerbelli-Gauthier, and Naomi Tanabe.

#### Nathan Conrad Jones (University of Illinois at Chicago)

Elliptic curve entanglements

## Résumé

#### Hershy Kisilevsky (Concordia University)

## Résumé

We consider heuristic predictions for small non-zero algebraic central values of twists of the L-function of an elliptic curve E/Q by Dirichlet characters. We provide computational evidence for these predictions and consequences of them for instances of an analogue of the Brauer-Siegel theorem.

#### Valeriya Kovaleva (Université de Montréal)

## Résumé

In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size $T^{3/2-\varepsilon}$. We will also explain how this result relates to Motohashi’s formula for the fourth moment, as well as the moments of moments of the Riemann Zeta and its maximum in short intervals.

#### Wanlin Li (Washington University in St.Louis)

## Résumé

Given an algebraic curve X of genus ≥3, one can construct two algebraic 1-cycles, the Ceresa cycle and the Gross-Kudla-Schoen modified diagonal cycle, each living in the Jacobian of X and the triple product X×X×X respectively. These two cycles are homologically trivial but are of infinite order in their corresponding Chow groups for a very general curve over C. From the work of S-W Zhang, for a fixed curve these two cycles are non-torsion in their corresponding Chow groups if and only if one of them is. In this paper, we prove that the Ceresa and Gross-Kudla-Schoen cycles associated to a modular curve X are non-torsion in the corresponding Chow groups when X=Γ∖H for certain congruence subgroups Γ⊂SL2(Z). We obtain the result by studying a pullback formula for special divisors by the diagonal map X↪X×X.

**Patrick Meisner (Ericsson)**

Arithmetic Statistic in Function Fields

## Résumé

Montgomery’s pair correlation conjecture first showed that there is a connection between zeros of L-functions and eigenvalues of random matrices. Katz and Sarnak then proved this to be the case in many cases over function fields in the q limit. In this talk, we will discuss lower order terms and their potential connections to random matrices.

#### Ram Murty (Queen’s University)

## Résumé

If F is an arithmetical function of period N, one can associate to F an L-function L(s,F) in the obvious way. If F(x) is +1 or -1 for x unequal to N and F(N)=0, then a famous conjecture of Erdos predicts that L(1,F) is never zero. We will discuss this conjecture and report on recent work with Abhishek Bharadwaj and Siddhi Pathak.

#### Francesco Pappalardi (Università degli Studi Roma Tre)

## Résumé

We will discuss a joint ongoing project with F. Campagna, N. Jones and P. Stevenhagen. It concerns points on elliptic curves over a number field with the property that the reduction of primes is never a generator for the reduced curve. Earlier work considered the case when the index of the reduction was always divisible by a fixed prime. The current project deals with other cases. The approach considers the affine representation modulo composite n on the 3 dimensional Z/nZ-vector space generated by the n–th roots of the points.

#### Jennifer Park (Ohio State University)

## Résumé

#### Lillian Pierce (Duke University)

## Résumé

In 1980 Carleson posed a question in PDE’s: how “well-behaved” must an initial data function be, to guarantee pointwise convergence of the solution of the linear Schrödinger equation? After nearly 40 years of study, this celebrated question was resolved by a combination of two results: one by Bourgain (2016), whose counterexample construction proved a necessary condition, and a complementary result of Du and Zhang (2019), who proved a sufficient condition. Bourgain’s counterexample was interesting for two reasons: first, it generated a necessary condition that contradicted what everyone had expected, and second, it was a number-theoretic argument. In this talk, we will survey how number theoretic ideas motivated Bourgain’s counterexample, and then see how they can be pushed much farther, in recent joint work with Rena Chu.

#### Maksym Radziwill (Northwestern University)

## Résumé

#### Zeév Rudnick (Tel Aviv University)

## Résumé

#### Arul Shankar (University of Toronto)

## Résumé

#### Nina Snaith (University of Bristol)

## Résumé

#### Lola Thompson (Utrecht University)

## Résumé

Let $s(n)$ denote the sum of proper divisors of an integer $n$. The function $s(n)$ has been studied for thousands of years, due to its connection with the perfect numbers. In 1992, Erd\H{o}s, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. This has been confirmed for certain specific sets $\mathcal{A}$, but remains open in general. In this talk, we will discuss some recent progress towards the EGPS conjecture. In particular, we show that the conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. This talk is based on joint work with K\ »{u}bra Benli, Giulia Cesana, C\'{e}cile Dartyge, and Charlotte Dombrowsky.

#### Caroline Turnage-Butterbaugh (Carleton College)

## Résumé

Assuming the Riemann Hypothesis, Montgomery (1973) proved a theorem concerning the pair correlation of nontrivial zeros of the Riemann zeta-function. One consequence of this theorem was that, under the Riemann Hypothesis, at least 2/3 of the zeros are simple. We show that this theorem of Montgomery holds unconditionally. Furthermore, under a hypothesis that is much weaker than RH, we show that at least 61.7% of zeros of the Riemann zeta-function are simple. This weaker hypothesis does not require that any of the zeros are on the half-line. This is joint work with Sieg Baluyot, Dan Goldston, and Ade Irma Suriajaya.

#### Tian Wang (MPIM/Concordia University)

Congruence class biases for cyclicity and Koblitz’s conjectures for elliptic curves

## Résumé

Let E be an elliptic curve over the rationals. In 1976, assuming GRH, Serre proved an analog of Artin’s primitive root conjecture for E, known as the cyclicity conjecture. A related conjecture, raised by Koblitz in 1988 and later refined by Zywina in 2011, has also drawn considerable attention. Recently, there has been increasing interest in the cyclicity conjecture for primes in arithmetic progression, with related work by Akbal-Guloglu and Wong. Building on Zywina’s argument for the refined Koblitz’s conjecture, we propose its variants for primes in arithmetic progression and provide theoretical and numerical evidence supporting its validity. In the talk, we will also show unexpected opposing average congruence class biases for cyclicity and Koblitz’s conjectures. This is joint work with Sung Min Lee and Jacob Mayle.