Aritmètica de varietats abelianes i aplicacions a matemètica discreta.

This project pretends to solve, using geometric techniques, some arithmetic questions about abelian varieties over number fields amd over finite fields, with a view to its application to the resolution of diophantine equations and to some topics of discrete matemathics. The concrete aims are: explicit caracterization of the possible torsion subgroups of an abelian variety with complex multiplication and low dimension, detemination of the formal group of the Nerón model of a semiabelian variety, construction of rigid analytic intermediate jacobians, explicit formulae for the cardinal of several moduli spaces over finite fields (hypereliptic curves, curves of genus 3 and MDS codes) and the development of effective algorithms for the law group of the jacobian of families of curves of genus 3, with a view to the construction of safer public key criptosystems.