By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola
This quantity comprises the improved types of the lectures given via the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are wide surveys of the present learn within the mathematics of elliptic curves, and likewise include a number of new effects which can't be chanced on in different places within the literature. due to readability and style of exposition, and to the heritage fabric explicitly incorporated within the textual content or quoted within the references, the amount is easily suited for examine scholars in addition to to senior mathematicians.
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Extra resources for Arithmetic theory of elliptic curves: lectures given at the 3rd session of the Centro internazionale matematico estivo
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. For arbitrary F, the situation seems more complicated. We had believed that this conjecture should continue to be valid, but counterexamples have recently been found by Michael Drinen. 11. Let E be an elliptic curve defined over $. Assume that SelE($,), is A-cotorsion. Then there exists a $-isogenous elliptic curve E' such that p ~ = t 0.
G$)) = -ordU(jE), where denotes the Tate period for E at v. Now a,, is clearly surjective. Hence ( ker(r,, ) 1 = [ ker(a,,) 1 . I ker(b,,) 1. 5, we have I ker(a,,)I = ~,@(f,,),l. 2, just the boundedness of J ker(a,,)l (and of I ker(b,,)J) suffices. To study ker(b,,) we use the following commutative diagram. c I I I 1 I , I 1 I I1 The surjectivity of the first row follows from Poitou-Tate Duality, which gives H2(M,C,) = 0 for any finite extension M of F,. )Thus, ker(bvn) 2 ker(dun). u, where u is a unit of F, and nu is a uniformizing parameter.
Im(n,) = Im(A,)di,. I I I1 1 1 Proof. The idea is quite simple. We know that Im(n,) and Im(X,) are p primary groups, that Irn(n,) is divisible, and has Z,-corank [M, : Q,]. It suffices to prove two things: (i) Im(n,) C Im(A,) and (ii) Im(A,) has Z,corank equal to [M, : Q,]. To prove (i), let c E Im(n,). We show that c E ker(H1(M,, E[pm]) t H'(M,, k[pw])), which coincides with Im(A,). If b E E(F,), we let % E Z(7,) denote its reduction. Let 4 be a cocycle representing c. Then +(g) = g(b)- b for all g E GMq, where b E ~ ( 7 ~ ) .