Geometric quadratic chabauty and other topics in number theory

Abstract

This thesis is is made of three parts.

The first part describes a generalization of the Chabauty's method, that can be used to determine the rational points of a curve such that s+g>r+1, where g is the genusof the curve, r is the rank of the Mordell-Weil group of the jacobian of the curve and s is the rank of the Neron-severi of the jacobian of the curve.

The second part proves that for all but a finite number of Cartan modular curves, the automorphisms are only the "expected" ones.

The last part describes an algorithm that solves in quasi-polynomial time the discrete logarithm problem in all finite fields whose characteristic is "small" with respect to the cardinality.