On the mordell - Weil rank of hyperelliptic curves

Abstract

The Jacobian of an algebraic curve C defned over a number feld K is an abelian variety whose group of K-rational points has finite rank, by the Mordell-Weil theorem. We study the proof of this theorem and the problem of computing the rank. We give some bounds on this rank in a certain family of curves by studying the Selmer group, which is a cohomological device that can be explicitly computed.