Aurélien Galateau

 Aurélien Galateau (Université Paris-Sud, Париж, Франция) посетил Лабораторию в октябре-ноябре 2012 г., а также в июне 2013 г.

В период с 29 октября по 30 ноября 2012 года он прочел курс лекций "Elliptic curves and Serre's "big image" theorem" для студентов и аспирантов факультета математики и стажеров-исследователей Лаборатории.

Abstract: The course will be an introduction to the theory of elliptic curves, with a view toward Serre's work on the Galois representations attached to their torsion. We will start with a geometric description (projective embedding, group law, isogenies, endomorphism ring) and then focus on the case of elliptic curves defined over finite fields (Hasse's estimate, ordinary and supersingular curves) and local fields (reduction, Néron-Ogg-Shafarevich criterion). Next, we will explain the link between complex multiplication (CM) on elliptic curves and class field theory of imaginary quadratic fields. The last part of the course will be devoted to non-CM elliptic curves and the Galois properties of their torsion points.

16 ноября 2012 г.  Aurélien Galateau выступил на еженедельном семинаре Лаборатории с докладом "Small points in subvarieties of abelian varieties".

Abstract: This talk will focus on a theorem of Ullmo and Zhang (the former Bogomolov conjecture) about the repartition of points of small height in subvarieties of abelian varieties. It is possible to give an 'effective' bound for small points under Serre's conjecture that primes of ordinary reduction for an abelian variety have density one. Further, one gets an unconditionnal 'effective' bound in the case of hypersurfaces. Both proofs use p-adic estimates on torsion points of abelian varieties and diophantine approximation.