On the size of generators of solutions of some diophantine equations

We will discuss analogies between the group of units of a number field (e.g. integral solutions of the equation $x^2-dy^2=1$) and the group of rational points of an abelian variety over a global field (e.g. rational solutions of the equation $y^2=x^3+ax+b$). Both groups are finitely generated and there is a natural notion of size or height, so the central question is to estimate the minimal size of a set of generators.