By J. Coates

This quantity comprises the increased models of the lectures given by way of the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers amassed listed below are vast surveys of the present study within the mathematics of elliptic curves, and likewise include numerous new effects which can't be stumbled on in different places within the literature. because 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 fitted to study scholars in addition to to senior mathematicians.

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**Example text**

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).