Euler characteristics and elliptic curves II

Abstract

This paper describes a generalisation of the methods of Iwasawa Theory to the field $F_{\infty }$ 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 $G_{\infty }$ denote the Galois group of $F_{\infty }$ over $F$. Then the main focus of this paper is on the study of the $G_{\infty }$-cohomology of the $p^{\infty }\text{-}$ Selmer group of $E$ over $F_{\infty }$, and the calculation of its Euler characteristic, where possible. The paper also describes proposed natural analogues to this situation of the classical Iwasawa $\lambda$-invariant and the condition of having $\mu$-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 $p=5$.