Variété abélienne | EPFL Graph Search

En mathématiques, et en particulier, en géométrie algébrique et géométrie complexe, une variété abélienne A est une variété algébrique projective qui est un groupe algébrique. La condition de est l'équivalent de la compacité pour les variétés différentielles ou analytiques, et donne une certaine rigidité à la structure. C'est un objet central en géométrie arithmétique. Définition Une variété abélienne sur un corps k est un groupe algébrique A sur k, dont la variété algébrique sous-jacente est projective, connexe et géométriquement réduite. Cette dernière condition veut dire que lorsque l'on étend le corps de base k à une clôture algébrique de k, la nouvelle variété est réduite (cela implique que A est réduite). Si k est de caractéristique nulle, la condition "géométriquement réduite" est automatiquement satisfaite pour tout groupe algébrique sur k (Théorème de Cartier). Exemple : Les variétés abéliennes de dimension 1 sont les courbes elliptiques. La jacobienne d'une

Capacities, Systoles and Jacobians of Riemann Surfaces

Björn Mützel

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety (PPAV) of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. The aim of this thesis is to contribute to the Schottky problem, which is to discern the Jacobians among the PPAVs. Buser and Sarnak approached this problem by means of a geometric invariant, the first successive minimum. They showed that the square of the first successive minimum, the squared norm of the shortest non-zero vector, in the lattice of a Jacobian of a Riemann surface of genus g is bounded from above by log(4g), whereas it can be of order g for the lattice of a PPAV of dimension g. The main goal of this work was to improve this result and to get insight into the connection between the geometry of a compact Riemann surface that is given in hyperbolic geometric terms, and the geometry of its Jacobian. We show the following general findings: For a hyperelliptic surface the first successive minimum is bounded from above by a universal constant. The square of the second successive minimum of the Jacobian of a Riemann surface of genus g is equally of order log(g). We provide refined upper bounds on the consecutive successive minima if the surface contains several disjoint small simple closed geodesics and a lower bound for the norm of certain lattice vectors of the Jacobian, if the surface contains small non-separating simple closed geodesics. If the concrete geometry of the Riemann surface is known, more precise statements can be made. In this case we obtain theoretical and practical estimates on all entries of the period Gram matrix. Here we establish upper and lower bounds based on the geometry of the cut locus of simple closed geodesics and also on the geometry of Q-pieces. In addition the following two results have been obtained: First, an improved lower bound for the maximum value of the norm of the shortest non-zero lattice vector among all PPAVs in even dimensions. This follows from an averaging method from the geometry of numbers applied to a family of symmetric PPAVs. Second, a new proof for a lower bound on the number of homotopically distinct geodesic loops, whose length is smaller than a fixed constant. This lower bound applies not only to geodesic loops on Riemann surfaces, but on arbitrary manifolds of non-positive curvature.

EPFL2011

Two variants of the support problem for products of abelian varieties and tori

Antonella Perucca

Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism phi of G and a non-zero integer c such that phi(P) = cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem). (C) 2009 Elsevier Inc. All rights reserved.

2009

On the Reduction of Points on Abelian Varieties and Tori

Antonella Perucca

Let G be the product of an abelian variety and a torus defined over a number field K. Let R-1, ..., R-n be points in G(K). Let l be a rational prime, and let a(1), ..., a(n) be nonnegative integers. Consider the set of primes p of K satisfying the following condition: the l-adic valuation of the order of (R-i mod p) equals a(i) for every i = 1, ..., n. We show that this set is either finite or has a positive natural density. We characterize the n-tuples a(1), ..., a(n) for which the density is positive. More generally, we study the l-part of the reduction of the points.

2011

Capacities, Systoles and Jacobians of Riemann Surfaces

Björn Mützel

EPFL2011