11/8/2004

Karl Rubin
Univ. California at Irvine

Elliptic curves and skew-Hermitian matrices

Suppose E is an elliptic curve defined over a number field K, and p is a prime where E has good ordinary reduction. The usual methods of Iwasawa theory give a single Iwasawa module from which one can recover the Selmer groups of E over all finite extensions of K in a Z_p^d-extension. These Selmer groups come with important parings: the Cassels pairing on the Tate-Shafarevich group and the p-adic height pairing on the Mordell-Weil group. In joint work with Barry Mazur we show that (under mild hypotheses) there is a free Iwasawa module with a skew-Hermitian pairing (unique up to isomorphism) from which one can recover not only the Selmer groups over all intermediate fields but all of the pairings as well. In this talk I will explain this result and give some examples.