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Assume that SelE(F,), is A-cotorsion. Then the characteristic ideal of XE(Fm) is fied by the involution L of A induced by ~ ( y = ) y-' for all y E r. A proof of this result can be found in [Gr2] using the Duality Theorems of Poitou and Tate. There it is dealt with in a much more general context-that of Selmer groups attached to "ordinary" padic representations. 2 completely in the following two sections. Our approach is quite different than the approach in Mazur's article and in Manin's more elementary expository article.

Silverman, The arithmetic of elliptic curves, Graduate Texts in Math. 106 (1986), Springer. [33] K. Wingberg, On Poincare' groups, J. London Math. Soc. 33 (1986), 271-278. Iwasawa Theory for Elliptic Curves Ralph Greenberg University of Washington 1. Introduction The topics that we will discuss have their origin in Mazur's synthesis of the theory of elliptic curves and Iwasawa's theory of Pp-extensions in the early 1970s. We first recall some results from Iwasawa's theory. Suppose that F is a finite extension of $ and that F, is a Galois extension of F such that Gal(F,/F) 2 Z,, the additive group of p a d i c integers, where p is any prime.

McConnell, Iwasawa theory of modular elliptic curves of analytic rank at most 1, J. London Math. Soc. 50 (1994), 243-264. J. Coates, R. Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129-174. J. Coates, S. Howson, Euler characteristics and elliptic curves, Proc. Nat. Acad. Sci. USA 94 (1997), 11115-11117. J. Coates, S. Howson, Euler characteristics and elliptic curves 11, in preparation. J. Coates, R. Sujatha, Galois cohomology of elliptic curves, Lecture Notes at the Tata Institute of Fundamental Research, Bombay (to appear).

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