Abstract

Let E and A be two elliptic curves, both defined over Q, and p>3 be a good ordinary prime for E. Assume that A has no complex multiplication so that the Galois group G of the extension F_\infty/Q=Q(A[p^{\infty}])/Q is an open subgroup of GL_2(Z_p). The aim of the talk is to investigate the dual Selmer group of E over F_\infty. Under certain technical hypotheses we prove that its characteristic element satisfies a functional equation. Assume further that there exists a prime q (different from p) such that (i) A has potentially multiplicative reduction at q and (ii) all the p-power division points of E are defined over the completion of F_\infty at a prime above q. As a consequence we show that the dual Selmer X(E/F_\infty) cannot be annihilated by any element in the centre of \Lambda(G). In particular, if in addition the \Lambda(H)-rank of X(E/F_\infty) equals 1 then X(E/F_\infty) is completely faithful. Unfortunately, this latter condition is never satisfied if E=A, but we do have examples of completely faithful Selmer groups when E is different from A. This is joint work in progress with T. Backhausz.