### 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>