On the size of generators of solutions of some Diophantine equations

It has been known since at least Fermat that the set of integral solutions to the equation $x^2-dy^2=1$ form a finitely generated group of rank one. It has been known since at least Poincaré that the set of rational solutions to equations of the type $y^2=x^3+ax+b$ form a group; in fact, as Mordell proved, the latter group is also finitely generated.
There is a natural notion of size or height of solutions, so an important and natural question is to estimate the minimal size of a set of generators. The questions can easily be generalized on one hand to the group of units of a number field and, on the other hand, to the group of rational points of an abelian variety over a global field.
The answer for the first case is essentially known, though there are important unsettled related questions; the answer for the second case is essentially conjectural. We will discuss what we know, conjecture and give examples where theorems may be proven. This will take us to a journey through some arithmetic geometry, zeta functions etc., i.e. several number theorists favourite toys.