Selmer groups and Mordell-Weil groups of elliptic curves over towers of function fields

In [12] and [13], Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers by a different method, allowing non-abelian Galois groups and removing the dependence on Tate’s conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of function fields whose Galois groups are p-adic Lie groups; a natural question is whether Mordell-Weil rank is bounded in such a tower. We give some Galois-theoretic criteria which guarantee that certain curves E/Q(t) have finite Mordell-Weil rank over C(tp), and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.