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

This quantity includes the elevated models of the lectures given by means of the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers amassed listed below are huge surveys of the present learn within the mathematics of elliptic curves, and likewise include a number of new effects which can't be stumbled on somewhere else within the literature. because of readability and magnificence of exposition, and to the heritage fabric explicitly incorporated within the textual content or quoted within the references, the amount is easily fitted to examine scholars in addition to to senior mathematicians.

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We have attempted to give a rather self-contained "Iwasawa-theoretic" approach to studying the above local Galois cohomology group. 2. But using results of Poitou and Tate is often Let T easier and more effective. We will illustrate this. Let C = ($,/Z,)($). denote its Tate module and V = T @zpQ,. The Z,-corank of H1(GM,,,C ) is just d i r n Q p ( H 1 ( ~ ,V)). , (Cocycles are required to be continuous. V has its $,-vector space topology. ) Letting hi denote dimQp(Hi(Mq,V)), then the Euler characteristic for V over M,, is given by c for any GM,,-representation space V.

We sketch an argument for (4). Let f (T) be a generator for the characteristic ideal of X , assuming that X is finitely generated and torsion over A. If we have f (C - 1) # 0 for all ppower roots of unity, then X/&X is finite for all n 0 and one estimates its order by studying f (C - l), where C runs over the pn-th roots of unity. One then could take a = X(f), b = p ( f ) in (4). Suppose X = A/(h(T)"), where h(T) is an irreducible element of A. If h(T) 8, for all n, then we are in the case just discussed.

12. Let E be an elliptic curve defined over F. Assume that SelE(F,), is A-cotorsion. The action of r = G a l ( F , / F ) on X E ( F ~ ) @ P ~ $ , is completely reducible. That is, ai = 1 for all i's such that f i ( T ) is not an associate of p. 59 Assume that E has good, ordinary reduction at all primes of F lying over p. 2 then holds. In particular, corankzp(sel~(F),),which is equal to rankap(XE (Fm)/TXE (F,)), would equal the power of T dividing ~ E ( T ) , would be equal to assuming the above conjecture.