By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

This quantity comprises the extended types of the lectures given by way of the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accumulated listed here are wide surveys of the present examine within the mathematics of elliptic curves, and in addition comprise a number of new effects which can't be came upon somewhere else within the literature. due to readability and magnificence of exposition, and to the heritage fabric explicitly incorporated within the textual content or quoted within the references, the amount is definitely fitted to study scholars in addition to to senior mathematicians.

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**Example text**

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.

Hence, in all cases, I ker(r,) 1 = c,(PI . Now assume that v lp. For each n, we let f u n denote the residue field for (F,),,. It doesn't depend on the choice of v,. Also, since v, is totally ramified in F,/F, for n >> 0, the finite field fun stabilizes to f,, the residue field of (F,),. We let denote the reduction of E at v. Then we have where y,, is a topological generator of Gal((F,),/(F,),,). Now E(f,), is finite and the kernel and cokernel of y,, - 1 have the same order, namely This is the order of ker(d,).

We then have a natural map 65 If rn, denotes the residue field of M,, then E[p"lG~V is just the pprimary subgroup of E(rn,), a finite group. Thus, ker(A,) is finite. The following lemma then suffices to prove (ii). If $ : GF, t Z,X is a continuous homomorphism, we will let (Q,/Z,)($) denote the group Q,/H, together with the action of GF, given by $. 3. H1(M,, (Q,/Z,)($)) has Zp-corank equal to [M, : Q,] 6, where 6 = 1 if is either the trivial character or the cyclotomic charGMV acter of GM,, and 6 = 0 otherwise.