Théorie d'Iwasawa p-adique locale et globale.
L’auteur présente des applications élémentaires de la théorie du corps de classes de Kato et Parshin en dimensions 1 et 3 : calcul du conducteur d’une extension de Witt-Artin-Schreier d’un corps local de dimension 1, et étude des revêtements abéliens des surfaces.
Nous établissons les résultats fondamentaux de la théorie -adique globale du corps de classes pour les corps de nombres.
On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.
1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function , Im z > 0, is a modular form of weight n/2 on , where N is the level of Q, i.e. is integral and has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly....
Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of -rational points of a curve of genus under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.