## Journal of the Mathematical Society of Japan

### 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 $p^{\infty}-$ 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 $p=5$.

#### Article information

Source
J. Math. Soc. Japan, Volume 53, Number 1 (2001), 175-235.

Dates
First available in Project Euclid: 9 June 2008

https://projecteuclid.org/euclid.jmsj/1213023976

Digital Object Identifier
doi:10.2969/jmsj/05310175

Mathematical Reviews number (MathSciNet)
MR1800527

Zentralblatt MATH identifier
1046.11079

#### Citation

COATES, John; HOWSON, Susan. Euler characteristics and elliptic curves II. J. Math. Soc. Japan 53 (2001), no. 1, 175--235. doi:10.2969/jmsj/05310175. https://projecteuclid.org/euclid.jmsj/1213023976