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

This quantity includes the accelerated 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 accrued listed below are huge surveys of the present study within the mathematics of elliptic curves, and in addition comprise a number of new effects which can't be came across 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 quantity is easily suited for examine scholars in addition to to senior mathematicians.

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**Example text**

That is, At/X r W, where W = M$ n pp-. On the other hand, if ppm M,, then XA-torsZ Zp(l), the Tate module for ppm. In this case, X/XA-torsis free and hence X Z At x Z p ( l ) . In the preceding discussion, the A-module At is in fact canonical. It is the reflexive hull of X/XA-t,,s. Thus, the action of A on X gives an action on At. Examining the above arguments more carefully, one finds that, for p odd, (One just studies the A-module X @for each At is isomorphic to A[A][~:QPI. character 4 of A. ) For p = 2, we can at least make such an identification up to a group of exponent 2.

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. It is easy to show that X/TX has Zp-rank 1. One must then show that X/T2X also has Zp-rank 1.

We refer the reader to [Be], [BeDal, 21, and [Maz4] for a discussion of this topic. 1. If q i p , then Im(n,) = 0. If qlp, then The first assertion can also be explained by using the fact that, for q p, H1(M,, E[pw]) is a finite group. But E(M,) 8 (Qp/Zp), and hence Im(tc,) are divisible groups. Even if M, is an infinite extension of Fv,it is clear from the above that Im(n,) = 0 if q i p. Assume that E has good, ordinary reduction at v, where v is a prime of F lying over p. Then, considering Eb*] as a subgroup of E ( F v ) , we have the reduction map E [ y ] t E[pm], where E is the reduction of E modulo v.