Résumés David 2024

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.

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.

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.

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.

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.

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.

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.

TBC

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.

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.

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.

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.