## Journal of the Mathematical Society of Japan

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

### 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 $p=5$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 9 June 2008

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/05310175

**Mathematical Reviews number (MathSciNet)**

MR1800527

**Zentralblatt MATH identifier**

1046.11079

**Subjects**

Primary: 11G05: Elliptic curves over global fields [See also 14H52] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

#### 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