Shioda described in his article [6] a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over kptq. In this article we find all elliptic curves over kptq for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.

In this paper, we consider the average size of the 2-Selmer groups of a class of quadratic twists of each elliptic curve over Q with Q-torsion group Z2 × Z2. We prove the existence of a positive proportion of quadratic twists of such a curve, each of which has rank 0 Mordell-Weil group.

The positive characteristic function-field Mordell-Lang conjecture for finite rank subgroups is resolved for curves as well as for subvarieties of semiabelian varieties defined over finite fields. In the latter case, the structure of the division points on such subvarieties is determined.

We study the quasi-endomorphism ring of certain infinitely definable subgroups in separably closed fields. Based on the results we obtain, we are able to prove a MordellLang theorem for Drinfeld modules of finite characteristic. Using specialization arguments we prove also a Mordell-Lang theorem for Drinfeld modules of generic characteristic.

We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of the affine space A over a p-adic field, endowed with polynomial actions on each coordinate of A. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.