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

# École diophantienne - Résumés

Francesco Amoroso (Université de Caen) : Small points on subvarieties of algebraic tori. Let G be a semi-abelian variety and V an irreducible algebraic subvariety of G, defined over the algebraic numbers. We assume that V is not a translate of a proper subtorus of G by a torsion point. Let h be a normalized height on the algebraic points of G. Thus the torsion points are the points of zero height. By the former Manin-Mumford conjecture, the set of torsion points of G lying on V is not Zariski dense. The former Bogomolov conjecture predicts more precisely that for some real $\theta > 0$, the set $V(\theta) := \{P \in V \mathrm{ \ such \ that \ } h(P) < \theta \}$ of « small points » is not Zariski dense in V. In these lessons we describe some quantitative versions of the Bogomolov conjecture for the (split) algebraic torus of dimension n, and we sketch how to prove these conjectures « up to an ε ». Notes du cours.

Pascal Autissier (Université Bordeaux 1) : Dynamical heights. In this course, I describe some arithmetic problems related to dynamical systems in the algebraic setting. I study polarized dynamical systems and explain the theory of canonical height functions. The theme is that many of the fundamental Diophantine conjectures have dynamical analogs. The end of the course is devoted to the equidistribution theorem of small points.

Maria Carrizosa (Université Lyon 1) : Around the Lehmer problem. We will start this course by recalling the classical Lehmer problem on algebraic numbers with small height. We will then study more thoroughly some of its generalizations : the relative Lehmer problem, and the case of points in some commutative algebraic groups of greater dimension. This problem is still open in general, but weaker versions can sometimes be proved, that happen to have useful applications in view of the Zilber-Pink conjecture. The end of the course will focus on these applications.

Philipp Habegger (Goethe Universität Frankfurt) : Small heights in big fields. The absolute logarithmic Weil height maps an algebraic number to a non-negative real. It vanishes precisely at zero and the roots of unity. In a given number field this height cannot be arbitrarily small without being zero. Such a height gap need not exist for an infinite algebraic extension of the rationals. However, in the 1970s Schinzel proved that the maximal totally real extension of the rationals admits a height gap. Bombieri and Zannier later proved the p-adic analog of this theorem. Amoroso, David, Dvornicich, Zannier and others, including the speaker, found height gaps under Galois-theoretic restrictions on the field. For example, Amoroso and Dvornicich proved that there’s a height gap in the maximal abelian extension of the rationals. To date, a characterization, even conjectural, of fields giving rise to a height gap remains an open problem. After giving a historical overview I will go into ideas that lead to proofs. The emphasis will be on Galois-theoretically restricted fields.

Marc Hindry (Université Paris 7) : An introduction to the theory of heights. The main theme of Diophantine Geometry is the description of rational or integral points on algebraic varieties in terms of the geometry of the latter. In elementary words it is the search of solutions in rational numbers for polynomial equations. The main tool is the so-called theory of heights. We will present a panorama of the various heights or "measures of arithmetic size" of points on an algebraic variety and a description of several main results and open problems in this field. Topics will include : heights à la Weil on projective spaces ; the "height machine" and applications ; heights à la Néron-Tate on elliptic curves and abelian varieties ; lower bounds for heights : problems of Lehmer, Lang and Bogomolov. The first hour will be a very elementary "Colloquium" talk ; the second hour will aim more at presenting tools and perspectives for the fives courses of the week.

Fabien Pazuki (Université Bordeaux 1) : The Lang-Silverman conjecture. The aim of this course is to study a conjecture of S. Lang about elliptic curves which was generalized by J. Silverman to the case of abelian varieties. It claims that for any number field k and any integer g, there exists $c(k,g)>0$ such that for any polarized abelian variety (A,D) defined over k of dimension g, for any k-rational point P that is non-torsion (and not inside an abelian subvariety) : $\hat{h}_{A,D}(P) \geq c(k,g) \mathrm{max} \{ 1, h_{Falt}(A/k) \}$, where $\hat{h}_{A,D}$ is the Néron-Tate height on (A, D) and $h_{Falt}$ is the Faltings height. We will study the main results obtained so far towards this conjecture and explain some of its many consequences in diophantine geometry.