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

This quantity comprises the elevated types of the lectures given through the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are extensive surveys of the present examine within the mathematics of elliptic curves, and in addition comprise numerous new effects which can't be came across in other places within the literature. as a result of readability and magnificence of exposition, and to the historical past fabric explicitly integrated within the textual content or quoted within the references, the quantity is easily fitted to learn scholars in addition to to senior mathematicians.

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**Example text**

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. $1 Remark. Because of the importance of this lemma, we will give a fairly selfcontained proof using local class field theory and techniques of Iwasawa Theory. But we then show how to obtain the same result as a simple application of the Duality theorems of Poitou and Tate.

If F = $, then it seems reasonable to make the following conjecture. For arbitrary F, the situation seems more complicated. We had believed that this conjecture should continue to be valid, but counterexamples have recently been found by Michael Drinen. 11. Let E be an elliptic curve defined over $. Assume that SelE($,), is A-cotorsion. Then there exists a $-isogenous elliptic curve E' such that p ~ = t 0. In particular, if Eb] is irreducible as a (ZIP+)representation of GQ,then p~ = 0. I I I Here E b ] = k e r ( ~ ( $ ) 3 E($)).

Then ker(rvn) is finite and has bounded order as n varies. If E has good reduction at v, then ker(rvn)= 0 for all n. Proof. Let Bv = H"(K, E[pm]), where K = (F,),. is unramified and finitely decomposed in F,/F, K is the unramified Z,extension of Fv (in fact, the only +,-extension of F,). The group B, is = isomorphic to (Qp/+p)e x (a finite group), where 0 5 e 5 2. Let run Gal(K/(F,),,,), which is isomorphic to H,, topologically generated by y,,,, say. Then + ker(g,) + coker(s,) + coker(h,). Therefore, we must study ker(h,), coker(h,), and ker(g,), which we do in a sequence of lemmas.