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

ISBN-10: 3540665463

ISBN-13: 9783540665465

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

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**Sample text**

In general, suppose that : El + E2 is an F-isogeny, where E l , E2 are defined over F. Let @ : SelEl (F,), + SelE,(F,), denote the induced A-module homomorphism. It is not hard to show that the kernel and cokernel , of @ have finite exponent, dividing the exponent of ker(+). Thus, S e l ~(F,), and SelE2(F,), have the same A-corank. If they are A-cotorsion, then the Xinvariants are the same. The characteristic ideals of XE, (F,) and XE2(F,) differ only by multiplication by a power of p. If F = $, then it seems reasonable to make the following conjecture.

One should compare this result with the remarks made above concerning S,, and S2. We will discuss below the cases where E has either multiplicative or supersingular reduction at some primes 6f F lying over p. But first we state an important conjecture of Mazur. The case where E has multiplicative reduction at a prime v of F lying over P is somewhat analogous to the case where E has good, ordinary reduction at v . In both cases, the GF,-representation space Vp(E) = Tp(E) 8 $, has an unramified 1-dimensional quotient.

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. We first prove that, when E has good, ordinary or multiplicative reduction at primes over p, the pprimary subgroups of SelE(Fn) and of SelE(F,) have a very simple and elegant description.

### 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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