### Euler Characteristics and Elliptic Curves II, by John H. Coates and Susan Howson

This paper has now appeared in J. Math. Soc. Japan vol. 53, no. 1, 2001, 175-235, and so the preprint has been removed. Let $E$ be an elliptic curve defined over a number field $F$ and with no complex multiplication over the algebraic closure of $F$. Let $p$ be any prime number. Our aim is to consider a generalisation of the methods of Iwasawa Theory to the field $F_{\infty }$ obtained by adjoining to $F$ all the fields of definition of the $p$-power torsion points on $E$. By a celebrated theorem of Serre the Galois group, $G_{\infty }$, of $F_{\infty }$ over $F$ is a non-Abelian $p$-adic Lie group of dimension 4. This situation was first considered by M. Harris in his 1977 Harvard PhD thesis, but remains shroudedin mystery today. We hope that our fragmentary results provide evidence that there is a deep and interesting Iwasawa Theory to be discovered. We have largely concentrated on the study of the $G_{\infty }$-cohomology of the $p$-Selmer group of $E$ over $F_{\infty }$ and the calculation of the Euler characteristic when these cohomology groups are all finite. However, in section 6 we also propose and discuss what seems to be the natural analogue of the vanishing of the Iwasawa $\mu$-invariant in the classical theory.

John H. Coates and Susan Howson <J.H.Coates@dpmms.cam.ac.uk, showson@math.mit.edu>