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 Calois 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 E/Q(t) have finite Mordell-Weil rank over C(t(p-infinity)), and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.
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