Accueil > Les trimestres du LMB > Automne 2013 > Conférence diophantienne - Résumés

# Conférence diophantienne - Résumés

Yuri Bilu (Université Bordeaux 1) : On the exponential local-global principle.
Let $A(n)= b_1a_1^n+...+b_sa_s^n$ be a "power sum" with coefficients in the ring $R$ of algebraic numbers finitely generated over $Q$. Skolem conjectured that the equation $A(n)=0$ has a solution $n \in Z$ if and only if it has a solution modulo every non-zero ideal of $R$. I will speak on a recent work with Boris Bartolome and Florian Luca, where we prove this assuming that the multiplicative group generated by $a_1, \ldots, a_s$ is of rank $1$.

Vincent Bosser (Université de Caen) : Minoration de la hauteur canonique associée à un module de Drinfeld.
Le but de l’exposé est de donner un aperçu des résultats connus concernant les problèmes de minorations de la hauteur canonique associée à un module de Drinfeld (problème de Lehmer et propriété de Bogomolov). L’exposé devrait être accessible aux non spécialistes des modules de Drinfeld.

José Ignacio Burgos (ICMAT Madrid) : Equidistribution of small points on toric varieties.
As the culmination of work of many mathematicians, Yuan has obtained a very general equidistribution result for small points in arithmetic varieties. Roughly speaking Yuan’s theorem states that given a « very » small generic sequence of points, with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to the measure associated to the hermitian line bundle. Here very small means that the height of the points converges to the lower bound of the essential minimum given by Zhang inequalities. The existence of a very small generic sequence is a strong condition on the arithmetic variety because it implies that the essential minimum attains its lower bound. We will say that a sequence is small if the height of the points converges to the essential minimum. By definition every arithmetic variety contains small generic sequences. We show that for toric line bundles on toric varieties arithmetic Yuan’s theorem can be splitted in two parts. A) Given a small generic sequence of points, with respect to a positive hermitian line bundle, the associated sequence of measures converges weakly to a measure. B) If the sequence is very small, the limit measure agrees with the measure associated to the hermitian line bundle.

Sara Checcoli (Université de Grenoble) : On the properties of Northcott and Bogomolov for fields and groups.
I will present a result in collaboration with M. Widmer on certain realizations of profinite groups having the properties of Northcott and Bogomolov defined by Bombieri and Zannier.

Huayi Chen (Université de Grenoble) : Majoration effective de la fonction de Hilbert-Samuel arithmétique.
On explique comment les idées de la géométrie d’Arakelov birationnelle, notamment les systèmes linéaires gradués et filtrés, peuvent être utilisées dans l’estimation effective des fonctions de Hilbert-Samuel géométrique et arithmétique.

Sinnou David (Université Paris 6) : Corps sans points de petite hauteur.
Zannier et Bombieri ont introduit la notion de corps possédant la propriété de Bogomolov, en lien avec le problème de Lehmer. Après une description des premiers résultats connus, nous proposerons des caractérisations suffisantes, de nature Galoisienne pour assurer cette propriété indépendamment des réalisations ainsi que les résultats que nous avons obtenus (avec F. Amoroso et U. Zannier).

Luca Demangos (UNA Mexico) : T-modules and Manin-Mumford conjecture.
We approach the Manin-Mumford conjecture from the point of view of G. Anderson’s T-modules. We propose a statement for a particular class of T-modules which present similarities with abelian varieties. We propose for such objects a proof strategy based on the ideas of U. Zannier and J. Pila, which should allow to tranform the study of the torsion points to a problem of Diophantine Geometry.

Carlo Gasbarri (Université de Strasbourg) : Geometric height inequalities for general surfaces in abelian varieties.
La conjecture de Vojta prévoit une inégalité tres puissante entre la hauteur par rapport au fibré canonique d’un point algébrique d’une variété projective et son discriminant. Elle est largement ouverte même dans le cas des corps des fonctions. Je décrirai l’état d’avancement de cette conjecture dans le cas où la variété est une surface lisse contenue dans une variété abélienne (travail en commun avec D. Brotbeck).

Éric Gaudron (Université de Clermont-Ferrand) : Géométrie des nombres généralisée.
Nous définirons deux séries de constantes d’Hermite associées à une extension algébrique de Q et nous établirons quelques-unes de leurs propriétés. Il s’agit d’un travail en commun avec Gaël Rémond.

Ariyan Javanpeykar (Universiteit Leiden) : Arakelov invariants of Belyi curves.
We bound the Faltings height of a curve explicitly in terms of the Belyi degree. We present several applications ranging from Diophantine geometry to computational aspects of modular forms.

Peter Jossen (Université Paris 11) : Local-global principles for subgroups of semi-abelian varieties.
Let $A$ be a semiabelian variety defined over a number field $k$, and let $Z$ be a finitely generated subgroup of the group of rational points $A(k)$. If a rational point of $A$ belongs to $Z$, then it lies necessarily in the $p$-adic closure of $Z$ inside $A(k_p)$ for all places $p$ of $k$. Whether or not, conversely, a rational point belongs to $Z$ as soon as it is $p$-adically close to $Z$ for all $p$ is not clear. I will report on some cases where this converse statement has a positive answer, and explain how the general problem is linked to Galois representations and Tate-Shafarevich groups attached to 1-motives.

Lars Kühne (SNS Pisa) : An effective result of André-Oort type.
The André-Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety means an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. The AOC is an analogue of the classical Manin Mumford conjecture on the distribution of torsion points in abelian varieties. I will present a rarely known approach to the AOC that goes back to Yves André himself, before the model-theoretic proofs of the AOC in certain cases by the Pila-Wilkie Zannier approach, André presented the first non-trivial proof of the AOC in case of a product of two modular curves. In my talk, I discuss results in the style of André’s method, allowing to actually compute all special points in a non-special curve of a product of two modular curves and more.

Samuel Le Fourn (Université Bordeaux 1) : Galois representations of quadratic Q-curves.
In this talk, we will discuss Serre’s uniformity problem for quadratic Q-curves and the associated projective Galois representations, and sketch the proof of uniformity in the imaginary quadratic case.

Valéry Mahé (EPFL) : Prime terms in divisibility sequences.
Divisibility sequences are integer sequences $\left( B_n \right)_{n\in\mathbb{N}}$ arising from the theory of dynamical systems. They satisfy the condition $B_n$ divides $B_m$ whenever $n$ divides $m$. For some divisibility sequences we will study the density of prime indices $n$ for which $B_n$ is prime using Galois theory.

Amilcar Pacheco (UF Rio de Janeiro) : Points rationnels des courbes sur un corps de fonctions.
Soit $X$ une courbe non-isotriviale définie sur un corps de fonctions $K$ en une variable sur un corps fini. On obtient une borne supérieure pour le nombre des points $K$-rationnels de $X$ en termes d’invariants géométriques, i.e., de $X$ et $K$. Cette borne ne dépend pas du rang du groupe de Mordell-Weil de la variété jacobienne de $X$.

Pierre Parent (Université Bordeaux 1) : Rational points on modular curves : diophantine approaches.

Patrice Philippon (Université Paris 6) : Méthode de Mahler et théorie de Galois.
Je montrerai comment décrire les relations de dépendance algébrique entre les valeurs de fonctions solutions d’un système d’équations fonctionnelles de type Mahler, à l’aide de la théorie de Galois de ces systèmes.

Andrea Surroca (Universität Basel) : On some conjectures on the Mordell-Weil group and the Tate-Shafarevich group of an abelian variety.
For an elliptic curve defined over the rationals numbers, Lang suggested a conjectural upper bound for the Néron-Tate height of the elements of a basis of the torsion-free part of the Mordell-Weil group. On the other hand, Goldfeld and Szpiro suggested a conjectural bound for the order of the Tate-Shafarevich group of the elliptic curve, in terms of the conductor. Following the approach of Manin, relying on the Birch and Swinnerton-Dyer conjecture and the Hasse-Weil conjecture for the L-series of the curve, we study extensions of these two conjectures for abelian varieties of arbitrary dimension defined over an arbitrary number field. As an application, we will mention a joint work with Bosser on the elliptic analogue of Baker’s method.

Francesco Veneziano (Universität Göttingen) : Torsion-anomalous intersections.
Anomalous Intersections are a fairly recent framework introduced by Bombieri, Masser and Zannier, which comprises and generalises a vast body of problems and conjectures in Arithmetic Geometry. Let $V$ be a variety contained in a group variety $G$, which is usually taken to be an abelian variety or a torus. When intersecting $V$ with an algebraic subgroup $B$, if the intersection $V\cap B$ has a component of dimension strictly greater than "expected", then such a component is said to be torsion-anomalous. In analogy with many fundamental results in the field, there are conjectures giving geometrical conditions for the variety $V$ to have only finitely many (maximal) torsion-anomalous subvarieties. The formulation of these conjectures generalises famous problems such as the Manin-Mumford Conjecture and is related to the Mordell-Lang problem.

Evgeniy Zorin (University of York) : Some applications of Mahler’s method.
We present some new measures of algebraic independence found with Mahler’s method and discuss a few concrete examples.