By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola
This quantity includes the accelerated models 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 accrued listed below are huge surveys of the present learn within the mathematics of elliptic curves, and likewise include a number of new effects which can't be discovered in other 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 learn scholars in addition to to senior mathematicians.
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Additional resources for Arithmetic theory of elliptic curves: lectures given at the 3rd session of the Centro internazionale matematico estivo
Its Pontryagin dual XE(F,) is an abelian pro-p group, which we regard as a A-module. It is not hard to prove that XE(F,) is finitely generated as a A-module (and so, SelE(F,), is a "cofinitely generated" A-module). 2. For XE(F) = Horn(Sel~(F),,$,/Z,) is known to be finitely generated over 7Zp. ) Write X = XE(F,) for brevity. 2, X/TX is finitely generated over Z,. Hence, X/mX is finite, where m = (p,T) is the maximal ideal of A. By a version of Nakayama's Lemma (valid for profinite A-modules X ) , it follows that XE(F,) is indeed finitely generated as a A-module.
Cp) for all n 2 0. B,, might be infinite. In fact, (Bu)div= pp- if pp C F,; (Bu)div = 0 if p, F,. Finally, assume that E has nonsplit, multiplicative reduction at v. Then c, = 1 or 2, depending on whether ord,(js) is odd or even. Using the Tate parametrization, one can see that B, is divisible when p is odd (and then ker(r,) = 0). If p = 2, E will have split, multiplicative reduction over K and so again B,/(B,)div has order related to ord,(&)). But 7, acts by -1 on this quotient. Hence ~ ' ( rB,) , , has order 1 or 2, depending on the parity of ord, (9;)).
Ochi, Ph. D. thesis, Cambridge 1999. -P. Serre, Cohomologie Galoisienne, Springer Lecture Notes 5, 5th edition (l994), Springer. -P. Serre, Proprittb galoisiennes des points d'ordre fini des courbes elliptiques Invent. Math. 1 5 (1972), 259-331. -P. Serre, Abelian L-adic representations, 1968 Benjamin. -P. Serre, Sur la dimension cohomologaque des groupes profinis, Topology 3 . (1965), 413-420. -P. Serre, Sur les groupes de congruence des variitis abiliennes I, 11, Izv. Akad. Nauk SSSR,, 28 (1964), .