Elliptic Curves and Skew-Hermitian Matrices

JOINT PRINCETON-IAS NUMBER THEORY SEMINAR
Topic:Elliptic Curves and Skew-Hermitian Matrices
Speaker:Karl Rubin
Affiliation:University of California, Irvine
Date:Monday, November 8
Time/Room:4:30pm - 5:30pm/Fine Hall 322

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.