Research Seminar Number Theory and Arithmetic Geometry

New cases of the generalised Brauer-Siegel theorem

Given an infinite sequence S of number fields we wonder about the asymptotic behavior, as K runs through S, of the product of the class number of K by its regulator of units (in terms of the discriminant of K). The classical Brauer-Siegel theorem answers this question when the number fields in S have bounded degree. In the early 2000's, Tsfasman and Vlăduţ suggested a conjecture which answers the question for much more general sequences. Their conjecture, which we call GBS, would follow from GRH, and is known to hold in a handful of situations : for instance, when the number fields in the sequence are almost normal over Q (Tsfasman, Vlăduţ, Zykin and Lebacque 2007).

In this talk I will discuss a work (in progress) with Philippe Lebacque and Gaël Rémond, in which we prove that GBS unconditionally holds in many new situations. Our main result deals with sequence of number fields with `small Galois complexity'. I will sketch the main ideas of our proof and, if time permits, I will exhibit a few concrete examples for which GBS is now proved.

Limits of Mahler measures and successively exact polynomials

The Mahler measure is an invariant which detects the arithmetic complexity of a polynomial. A conjecture of Boyd predicts that the set of real numbers which can be expressed as the Mahler measure of a polynomial with integer coefficients (in any number of variables) should be closed. If this was true, such a set could not accumulate towards its minimum, which would answer a celebrated conjecture asked by Lehmer exactly ninety years ago. In this talk we will show how one can construct some natural sequences of polynomials whose Mahler measures converge to the Mahler measure of some other polynomial, basing our exposition on joint work with François Brunault, Antonin Guilloux and Mahya Mehrabdollahei. Moreover, we will show how to bound explicitly the speed of convergence in these limits, and how to give complete asymptotic expansions in some explicit examples. In some cases, these examples are provided by sequences of so-called exact polynomials. If time permits, we will conclude our talk by explaining a generalization of this definition, given in joint work with François Brunault, which allows us to predict several new connections between Mahler measures and special values of L-functions.

Algebraicity of Higher Green Functions at a CM Point

By the classical theory of complex multiplication, the modular j-function takes algebraic values at CM points. It is an interesting question to ask about the algebraic nature of other types of automorphic functions at CM points. For the automorphic Green function at integral parameters, Gross and Zagier conjectured in the 1980s that their values at a CM point is essentially the logarithm of an algebraic number. In this talk, we will discuss recent progress toward and generalization of this conjecture. This is partly joint with Jan Bruinier and Tonghai Yang.

Arithmetic sparsity in mixed Hodge settings

The study of equations and their rational solutions is one of the oldest and hardest subjects in mathematics. Landmark results in this area were the finiteness theorems by Faltings. For higher dimensional varieties of general type, non-density or sparsity are expected, and finiteness is expected only in very specific cases. We will first give a brief overview of previous results and conjectures on finiteness, non-density, and sparsity of integral/rational points, especially those on moduli spaces. After a review of the notion of variation of mixed Hodge structures, we will present new sparsity results on integral points in the mixed Hodge setting, and discuss possible future directions.

3-torsion of abelian surfaces and generalized Kummer fourfolds

An abelian variety and its dual are derived equivalent, but it is an open question whether generalized Kummer varieties attached to an abelian surface and its dual are derived equivalent. In recent work joint with Honigs and Voight, we produced examples of abelian surfaces defined over Q whose associated Kummer fourfold is not derived equivalent over Q to that of its dual. In this talk, I will discuss how we construct examples of abelian surfaces whose 3-torsion is not isomorphic to that of its dual, and how this relates to the non-derived equivalences.