Arithmetic Geometry Year

Poncelet French-Russian Laboratory,

Moscow, Russia

2012 - 2013

Seminar Program

There is a weekly arithmetic seminar taking place on Wesdnesday 17:30-19:00 at the Independent Universty of Moscow (11, Bolshoi Vlassievsky pereulok), room 304. The next talks:

The previous talks:

20/03/2013 Arnaud Durand (Université Paris-Sud, France)
Metric Diophantine approximation on the middle-third Cantor set

Let M(μ) be the set of all real numbers that are approximable at a rate at least a given μ≥ 2 by the rationals. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of M(μ) is equal to 2/μ. We consider the further question of the size of the intersection of M(μ) with Ahlfors regular compact subsets of the interval [0,1]. In particular, we propose a conjecture for the exact value of the dimension of M(μ) intersected with the middle-third Cantor set. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. This study relies on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points. This is a joint work with Yann Bugeaud (Strasbourg).

24/04/2013 Yuri Bilu (Université de Bordeaux 1)
Effective proofs of Andre theorem on complex multiplication points on curves

20/03/2013 Philippe Lebacque (LM Besançon, Poncelet)
On the cohomological dimension of some pro-p-extensions

This talk deals with pro-p-extensions of number fields and function fields with cohomological dimension at most 2. I will first recall some old results concerning Galois groups of type G_S(p) , then I will discuss the method introduced by Labute and Schmidt to control the case "S finite" and give some corollaries concerning Tsfasman-Vladuts φ_q's, and I will conclude giving new situations for number fields with cd_pG≤2 (joint work with Blondeau and Maire).

06-13/03/2013 Marc Hindry (IM Jussieu-Laboratoire Poncelet)
On the size of generators of solutions of some Diophantine equations

We will discuss analogies between the group of units of a number field (e.g. integral solutions of the equation x^2-dy^2=1) and the group of rational points of an abelian variety over a global field (e.g. rational solutions of the equation y^2=x^3+ax+b). Both groups are finitely generated and there is a natural notion of size or height, so the central question is to estimate the minimal size of a set of generators.

Poncelet Laboratory home page

Sponsors: ANR, Campus France, CNRS, french ambassy, HSE Lab. of Alg. Geom., IUM

20/03/2013	Arnaud Durand (Université Paris-Sud, France)
	Metric Diophantine approximation on the middle-third Cantor set
	Let M(μ) be the set of all real numbers that are approximable at a rate at least a given μ≥ 2 by the rationals. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of M(μ) is equal to 2/μ. We consider the further question of the size of the intersection of M(μ) with Ahlfors regular compact subsets of the interval [0,1]. In particular, we propose a conjecture for the exact value of the dimension of M(μ) intersected with the middle-third Cantor set. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. This study relies on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points. This is a joint work with Yann Bugeaud (Strasbourg).
24/04/2013	Yuri Bilu (Université de Bordeaux 1)
	Effective proofs of Andre theorem on complex multiplication points on curves
20/03/2013	Philippe Lebacque (LM Besançon, Poncelet)
	On the cohomological dimension of some pro-p-extensions
	This talk deals with pro-p-extensions of number fields and function fields with cohomological dimension at most 2. I will first recall some old results concerning Galois groups of type G_S(p) , then I will discuss the method introduced by Labute and Schmidt to control the case "S finite" and give some corollaries concerning Tsfasman-Vladuts φ_q's, and I will conclude giving new situations for number fields with cd_pG≤2 (joint work with Blondeau and Maire).
06-13/03/2013	Marc Hindry (IM Jussieu-Laboratoire Poncelet)
	On the size of generators of solutions of some Diophantine equations
	We will discuss analogies between the group of units of a number field (e.g. integral solutions of the equation x^2-dy^2=1) and the group of rational points of an abelian variety over a global field (e.g. rational solutions of the equation y^2=x^3+ax+b). Both groups are finitely generated and there is a natural notion of size or height, so the central question is to estimate the minimal size of a set of generators.