Euler characteristics and elliptic curves II

To the memory of Kenkichi Iwasawa

Abstract

This paper describes a generalisation of the methods of Iwasawa Theory to the field F_∞ obtained by adjoining the field of definition of all the p-power torsion points on an elliptic curve, E, to a number field, F. Everything considered is essentially well-known in the case E has complex multiplication, thus it is assumed throughout that E has no complex multiplication. Let G_∞ denote the Galois group of F_∞ over F. Then the main focus of this paper is on the study of the G_∞-cohomology of the p^∞-Selmer group of E over F_∞, and the calculation of its Euler characteristic, where possible. The paper also describes proposed natural analogues to this situation of the classical Iwasawa λ-invariant and the condition of having μ-invariant equal to 0.
The final section illustrates the general theory by a detailed discussion of the three elliptic curves of conductor 11, at the prime p=5.