By J. Coates

ISBN-10: 0203645375

ISBN-13: 9780203645376

This quantity includes the accelerated models of the lectures given via the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed here are huge surveys of the present examine within the mathematics of elliptic curves, and in addition include numerous new effects which can't be discovered in other places within the literature. as a result of readability and style of exposition, and to the historical past fabric explicitly integrated within the textual content or quoted within the references, the quantity is definitely suited for learn scholars in addition to to senior mathematicians.

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**Additional resources for Arithmetic Theory of Elliptic Curves: Lectures given at the Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetaro, Italy, ... Mathematics / Fondazione C.I.M.E., Firenze)**

**Sample text**

The invariants X and p can be obtained from 03 C ciTi, where ci E Z p for i=O i 2 0. Let p(f) 2 0 be defined by: p"(f)lf (T), but pp(f)+l I( f (T) in A. Thus, f ( ~ ) ~ - p ( is f )in A and has at least one coefficient in Z;. Define X(f) 2 0 to be the smallest i such that cip-~(f)E B;. The ideal (f (T)) of A is called a= 1 the "characteristic ideal" of X. Then it turns out that the X and p occurring in Iwasawa's theorem are given by X = X(f), p = p(f). , or f,(T) is an associate of a monic polynomial of degree X(f,), irreducible over $, and "distinguished" (which means that the nonleading coefficients are in pZp), as a group.

Otherwise, we find that H2(Mq,T) is finite and that which is a finite cyclic group, indeed isomorphic to HomcMq(+,(I), C). This argument works even for p = 2. We want to mention here one useful consequence of the above discussion. where $ : GF,, -+ Z; is a continuous homoAgain we let C = ($,/Z,)($), morphism, v is any prime of F lying over p. If 77 is a prime of F, lying over v, then (F,), is the cyclotomic Z,-extension of F,. 3, the Z,corank of H1((Fn),, C) differs from [(F,),, : Fv]by at most 1.

Let C denote the image of W,(A) in A. Then C $ , / Z p as a group. Here then is a definition of the ~ S e l m e group r SA($), for A over $: + 72 Iwasawa theory for elliptic curves Ralph Greenberg where v runs over all primes of $. Here we take L, = 0 for v # p, analogously to the elliptic curve case. One defines L, = Im(Xp)div,where is the natural map. In [Gr3], one can find a calculation of SA($),, and also SA($,),, for p = 11,23, and 691. One can make similar definitions whenever one has a padic Galois representation with suitable properties.

### Arithmetic Theory of Elliptic Curves: Lectures given at the Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetaro, Italy, ... Mathematics / Fondazione C.I.M.E., Firenze) by J. Coates

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