Elliptic Curves over Global Fields and ℓ-Adic Representations
In the previous two chapters the local study of elliptic curves was carried out and a substantial part of the theory was related to how the fundamental symmetry, the Frobenius element, behaved on the curve modulo a prime. For an elliptic curve E over a number field K (or more generally any global field), we have for each prime a Frobenius element acting on certain points of the curve. These Frobenius elements are in Gal(K s/K), and this Galois group acts on the K s-valued points E(K s) on the group N E = N E(K s), the subgroup of N-division points, and on the limit Tate modules T ℓ(E) where N and ℓ are prime to the characteristic of K. In Chapters 12 and 13 the action of Gal(K s/K) on the endomorphisms EndK s(E) and the automorphisms AutK s(E) over K s was considered in detail. As usual K s denotes a separable algebraic closure of K.
KeywordsElliptic Curve Elliptic Curf Galois Group Abelian Variety Number Field
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