Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Characteristic elements, pairings and functional equations over the false Tate curve extension

GERGELY ZÁBRÁDIa1

a1 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB.

Abstract

We construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a prime p≥5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we compute the characteristic elements of those modules – arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension – which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

(Received October 10 2006)

(Revised May 29 2007)