Journal of the Mathematical Society of Japan

Euler characteristics and elliptic curves II

John COATES and Susan HOWSON

Full-text: Open access


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.

Article information

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

First available in Project Euclid: 9 June 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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.

Export citation