Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, p is an odd prime, and L is a p-extension of K that is Galois over K. Let c be an element of order 2 in Gal(L/k), and H the subgroup of all elements of G := Gal(L/K) that commute with c. Under very mild hypotheses the Parity Conjecture (combined with a little representation theory) predicts that if the rank of E(K) is odd, then the rank of E(L) is at least [G:H]. For example, if L/k is dihedral and the rank of E(K) is odd, then the rank of E(L) should be at least [L:K].

In this talk I will discuss recent joint work with Barry Mazur, where we prove an analogue of this result with "rank" repaced by "p-Selmer rank".