Arithmetic theory of elliptic curves: lectures given at the by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

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

This quantity includes the multiplied models of the lectures given by means of the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed here are vast surveys of the present learn within the mathematics of elliptic curves, and likewise comprise numerous new effects which can't be stumbled on in other places within the literature. due to readability and magnificence of exposition, and to the historical past fabric explicitly integrated within the textual content or quoted within the references, the quantity is easily suited for learn scholars in addition to to senior mathematicians.

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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), . 3-20 and 35 (1971), 731-737.

Assume also that SelE(F), is finite. Then sel~(F,), is A-cotorsion. 2, using the following exercise: if X is a A-module such that X/TX is finite, then X is a torsion A-module. The hypothesis on SelE(F), is equivalent to assuming that both the Mordell-Weil group E ( F ) and the pshafarevich-Tate group IIIE(F), are finite. 3 is known is the following. The special case where E has complex multiplication had previously been settled by Rubin [Rul]. 5 (Kato-Rohrlich). Assume that E is defined over $ and is modular.

Suppose that SelE(F,), is A-cotorsion. Let fE(T) be a generator of the characteristic ideal of XE(F,). Then XE = X ( ~ E and ) p~ = p ( f ~ ) We . have where the fi(T)'s are irreducible elements of A, and the ai's are positive. If (fi(T)) = (p), then it is possible for ai > 1. However, in contrast, it seems reasonable to make the following "semi-simplicity7' conjecture. 12. Let E be an elliptic curve defined over F. Assume that SelE(F,), is A-cotorsion. The action of r = G a l ( F , / F ) on X E ( F ~ ) @ P ~ $ , is completely reducible.

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