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J. Coates, R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129-174. J. Coates, S. Howson, Euler characteristics and elliptic curves, Proc. Nat. Acad. Sci. USA 94 (1997), 11115-11117. J. Coates, S. Howson, Euler characteristics and elliptic curves 11, in preparation. J. Coates, R. Sujatha, Galois cohomology of elliptic curves, Lecture Notes at the Tata Institute of Fundamental Research, Bombay (to appear). J. Coates, R. Sujatha, Iwasawa theory of elliptic curves, to appear in Proc.

The ideal (f (T)) of A is called a= 1 the "characteristic ideal" of X. Then it turns out that the X and p occurring in Iwasawa's theorem are given by X = X(f), p = p(f). , or f,(T) is an associate of a monic polynomial of degree X(f,), irreducible over $, and "distinguished" (which means that the nonleading coefficients are in pZp), as a group. in which case p(f,) = 0 and A/(f,(T)a*) is isomorphic to Then, X = Ca,X(fi), p = Ca,p(f,). The invariant X can be described more simply as X = rankzp(X/Xmp-tors),where XzP-to,, is the torsion subgroup of X.

Thus, if we let rv= Gal((F,)q/F,), then it follows that as n -+ oo corankzp(HI ((F,)~, ~ ) )= ~ pn[Fv f : Q,] + O(1). Iwasawa theory for elliptic curves Ralph Greenberg 68 The structure theory of A-modules then implies that H1((F,),, C) has corank equal to [F, : $,I as a Z,[[r,]]-module. Assume that $ is unramified and that the maximal unrarnified extension of F, contains no p t h roots of unity. (If the ramification index e, for v over p is 5 p - 2, then this will be true. ) Then by (2) we see that H1(F,, C) is divisible.