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

ISBN-10: 3540665463

ISBN-13: 9783540665465

This quantity includes the increased types 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 accumulated listed here are vast surveys of the present study within the mathematics of elliptic curves, and likewise comprise a number of new effects which can't be came across in other places within the literature. as a result of readability and style of exposition, and to the heritage fabric explicitly integrated within the textual content or quoted within the references, the quantity is easily suited for study scholars in addition to to senior mathematicians.

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**Sample text**

Therefore, ker(r,,) = 0 as stated. c One can determine the precise order of ker(r,, ), where vn 1 v and v is any nonarchimedean prime of F not dividing p where E has bad reduction. This will be especially useful in section 4, where we will need ( ker(r,)l. The result is: I ker(r,)l = c,(PI , where cp) is the highest power of p dividiG the Tamagawa factor c, for E at v. Recall that c, = [E(F,): Eo(F,)], where Eo(F,) is the subgroup of local points which have nonsingular reduction at v. First we consider the case where E has additive reduction a t v.

Howson, Zwasawa theory of elliptic curves for p-odic Lie extensions, Ph. D. 1181 . thesis, Cambridge 1998. I191 K. Iwasawa, On &-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326. 1201 H. Imai. A remark on the rational points of abelian varieties with values in L--A cyclotoiic Zpextensions, Proc. ~ a Acad. ~ 5k1 (1975), 12-16. [21] S. Lang, H. Trotter, hbenius distributions in GLz-extensions, Springer Lecture Notes 504 (1976), Springer. [22] M. Lazard, Groupes analytiques p-adiques, Publ.

3, the Z,corank of H1((Fn),, C) differs from [(F,),, : Fv]by at most 1. Thus, if we let rv= Gal((F,)q/F,), then it follows that as n -+ oo corankzp(HI ((F,)~, ~ ) )= ~ pn[Fv f : Q,] + O(1). Iwasawa theory for elliptic curves Ralph Greenberg 68 The structure theory of A-modules then implies that H1((F,),, C) has corank equal to [F, : $,I as a Z,[[r,]]-module. Assume that $ is unramified and that the maximal unrarnified extension of F, contains no p t h roots of unity. (If the ramification index e, for v over p is 5 p - 2, then this will be true.

### Arithmetic theory of elliptic curves: lectures given at the 3rd session of the Centro internazionale matematico estivo by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

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