Une petite journée GANDA
IMB le 3 avril 2019
Résumés des exposés
Régulateurs des courbes elliptiques
Soit \(E\) une courbe elliptique sur un corps de nombres \(K\). Dans cet exposé, je
définis et étudie le régulateur de \(E\) sur \(K\), qui est un invariant arithmétique
associé au groupe de Mordell-Weil \(E(K)\). Plus précisément, je présente un
travail en commun avec Hindry et Pazuki, montrant une propriété de finitude
"à la Northcott" pour ce régulateur.
Cyclic reduction of elliptic curves
Let \(E\) be an elliptic curve defined over a number field \(K\) and \(p\) a prime of good reduction for \(E\). It is well known that the group of points on \(E\) modulo \(p\) is a finite abelian group on at most two generators. In this talk we will address the problem of determining the density \(\delta\) of the set of primes of \(K\) for which \(E\) has a cyclic reduction. This problem turns out to be very similar to Artin's primitive root conjecture, in the sense that it can be studied by looking at the primes in \(K\) that do not split completely in an infinite family of finite extensions of \(K\). Under GRH we can derive an explicit formula for \(\delta\). The possible vanishing of \(\delta\) depends non-trivially on the Galois representation associated to the torsion points of \(E\).
Arithmetic volume formulae for Siegel modular varieties
By a work of Siegel it is known that the volumes of Siegel
modular varieties can be expressed essentially in terms
of special values of the Riemann \(\zeta\)-function. In our
talk we will discuss an arithmetic analogue of Siegel's
volume formula, which has been proven for genus one
and two, and is conjectural for higher genera.
X-coordinates of Pell equations which are factorials
Let \(d>1\) be a squarefree integer and \((X_n,Y_n)\) be the \(n\)th solution of the Pell equation \(X^2-dY^2=\pm 1\). In my talk, I will show that
if \(X_n=m!\) for some positive integer \(m\), then \(n=1\). The proof uses ingredients from the proof of the Primitive Divisor Theorem for Lucas sequences
by Bilu, Hanrot and Voutier as well as recent results on distribution of primes in progressions with small moduli by Bennett, Martin, O'Bryant and Rechnitzer.
This is joint work with S. Laishram and M. Sias.
Théorème de Bertini et propriété de Northcott
Soit \(X\) une variété projective lisse sur un corps de nombres \(K\). Le but de l'exposé est d'expliquer comment tracer une courbe algébrique \(C\) sur \(X\), avec contrôle sur son genre, sur son degré, sur sa hauteur et sur son corps de définition. Lorsque \(X\) est une variété abélienne, cela permet de construire une machine à "majorer par la hauteur de Faltings", que nous ne nous priverons pas d'utiliser.
Mahler measure, reciprocality and L-functions
We will explore the connections between special values of \(L\)-functions and
the Mahler measure of polynomials, which started with the work of Boyd on Lehmer's problem for polynomials in multiple variables, and with its motivic interpretation due to Deninger.
Time permitting, we will report on our work in progress to construct a polynomial whose Mahler measure is related to a Dirichlet character using Beilinson's Eisenstein symbol in motivic cohomology.
Greatest common divisors of linear recurrences
The greatest common divisor of two linear recurrences is a simple object whose distribution possesses a wide range of behaviors according to the properties of the recurrences themselves. We shall briefly review what is classically known from the works of Bugeaud-Corvaja-Zannier, Fuchs, and others, on large values when the recurrences are non-degenerate, through Schmidt's Subspace Theorem, and Silverman's geometric reformulation connecting them to Vojta's conjecture. We shall then focus on recent progress, by the speaker and others, when one of the recurrences is degenerate: one can then find precise results for the distribution of the G.C.D.'s. The methods in this case are of a more analytic flavor and produce theorems concerning small values and averages instead. Those also constitute an interesting case study for unified conjectures.
Avec de soutien financier de l'IMB et de l'IRN CNRS « GANDA »
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