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

Abstract

Silverman has discussed the problem of bounding the Mordell–Weil ranks of elliptic curves over towers of function fields (J. Algebraic Geom. 9 (2000), 301–308; J. reine. angew. Math. 577 (2004), 153–169). We first prove generalizations of the theorems of Silverman 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 the Mordell–Weil rank is bounded in such a tower. We give some Galois-theoretic criteria which guarantee that certain curves ${\mathcal{E}}/{\mathbb{Q}}(t)$ have finite Mordell–Weil rank over ${\mathbb{C}}(t^{p^{-\infty}})$, and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.