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".

An explicit expression for the L-function evaluator associated to an abelian extension K/k of number fields of degree 2p will be discussed. Instances will be given where this expression can be used to determine pieces of the ideal class group of K that are annihilated by this L-function evaluator.

Given an arbitrary polynomial f in n variables over a finite field k, it is known that for a generic linear form l the exponential sum associated to f(x)+l(x) is pure. However, the proof is non-constructive and gives no explicit description of the set of such l's. In this talk we will give some results and conjectures related to the problem of giving an explicit geometric description of this set.