Suppose E is an elliptic curve defined over a number
field K, and p is a prime where E has good ordinary reduction.
We wish to study the Selmer groups of E over all finite extensions
L of K contained in the maximal Z_p-power extension of K, along
with their p-adic height pairings and a Cassels pairings.
Our goal is to produce a single free Iwasawa module of finite
rank, with a skew-Hermitian pairing, from which we can recover
all of this data. Using recent work of Nekovar we can show that
(under mild hypotheses) such an `organizing module' exists, and we
will give some examples.
This work is joint with Barry Mazur.