Karl Rubin

Growth of Selmer rank in nonabelian extensions of number fields

Suppose E is an elliptic curve defined over a number field k, K/k is a quadratic extension, and F is a Galois extension of k containing K. The Parity Conjecture gives a lower bound for the rank of E(F) that sometimes can be quite large. For example, if F/k is dihedral and the rank of E(K) is odd, then under mild assumptions the rank of E(F) should be at least [F:K]. In this talk I will discuss recent joint work with Barry Mazur, where we prove lower bounds of this type when p is an odd prime, F/K is a p-extension, and with "rank" repaced by "p-Selmer rank".