Rank of Elliptic Curves over Almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinite field K. Let K denote a separable closure of K, σ an element of the Galois group GK = Gal(K/K), and K(σ) the invariant subfield of K. We prove that if the characteristic of K is not 2 and σ belongs to a suitable open subgroup of GK , then E(K(σ)) has infinite rank. In [2], G. Frey and M. Jarden considered the rank of abelian varieties over fields which are almost separably closed. Specifically, let K denote a field of finite type, K a separable closure of K, and GK = Gal(K/K). For σ1, . . . , σn a sequence of elements of GK , let K(σ1, . . . , σn) denote the fixed field of the σi. According to [2], if A is an abelian variety over K, there is a subset of GK of measure 1 such that for every n-tuple belonging to the subset, A(K(σ1, . . . , σn)) has infinite rank. Question 1: Is is true that for every choice of K, A, n, and σi, dimA(K(σ1, . . . , σn))⊗Q =∞? This paper is intended to provide some evidence for the author’s suspicion that the answer to the above question is positive, at least when A is an elliptic curve. We mainly discuss the simplest case, dimA = n = 1. It should, perhaps, be remarked that it is known [3] that the torsion of A(K(σ1, . . . , σn)) need not be infinite. Of course, this is a rather different kind of problem in that the Galois module A(K)tor is finite dimensional, whereas A(K)⊗Q is infinite dimensional. * Partially supported by the Sloan Foundation and by NSF Grant DMS 97-27553. AMS Classification 11G05