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

This quantity comprises the increased types of the lectures given by means of the authors on the C. I. M. E. tutorial 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 likewise comprise numerous new effects which can't be discovered somewhere else 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 amount is definitely fitted to study scholars in addition to to senior mathematicians.

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**Example text**

Notice that, since @/Fis abelian, the action of T = y - 1 on ~al(k:/F,) is trivial. Thus, X/TX is infinite. Now if one considers the A-module Y = A/( fi (T)ai),where f i (T) is irreducible in A, then Y/TY is infinite if and only if fi(T) is an associate of T. Therefore, if F is an imaginary quadratic field in which p splits and if F, is the cyclotomic Bpextension of F, then TI f (T), where f (T) is a generator of the characteristic ideal of X . One can prove that T2 I( f (T). (This is an interesting exercise.

But 7, acts by -1 on this quotient. Hence ~ ' ( rB,) , , has order 1 or 2, depending on the parity of ord, (9;)). 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,),,).

Also, we have lqElv = ljElul. Let denote the reciprocity map of local class field theory. We will prove the following result. 6. Let M be a finite extension of F,. Then If M is a +,-extension of F,, then a~ is surjective. Proof. The last statement is clear since GM has pcohomological dimension 1 if M/F, has profinite degree divisible by p". For the first statement, the exact sequence induces a map aC):H1(M, E[pn]) + H1(M, Z/pnZ) for every n 2 1. Because of the Weil pairing, we have Hom(E[pn],p,- ) E E[pn].