Recent work of Nekovar makes it possible to produce families of examples of elliptic curves E and Z_p^d-extensions L/K such that the p-Selmer rank of E over F is at least [F:K] for every finite extension F of K in L. The well-known situation where K is imaginary quadratic, E is an elliptic curve over Q with odd rank over K, and L is the anticyclotomic Z_p-extension of K, is an example of this phenomenon.

In this talk I will discuss a proof of Nekovar's theorem using the ideas introduced in Barry Mazur's lecture. I will also discuss some root number calculations and use them to give concrete examples of large Selmer growth. This work is joint with Barry Mazur.