By Arthur Cayley

This quantity is made out of electronic photos from the Cornell college Library historic arithmetic Monographs assortment.

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This Elibron Classics version is a facsimile reprint of a 1905 variation by way of Macmillan and Co. , Ltd. , London.

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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.