Lectures on Shimura Curves 1: Endomorphisms of Elliptic Curves

0.1. Endomorphisms of elliptic curves. Recall that a homomorphism of complex elliptic curves is just a holomorphic map E1 → E2 which preserves the origin. (It turns out that this condition is enough to force it to be a homomorphism of groups in the usual sense; why?) An isogeny of elliptic curves is a homomorphism whose kernel is a finite subgroup of E1. In fact the kernel of a homomorphism of Lie groups is a closed Lie subgroup, so in this case is either finite or all of E1, so an isogeny of elliptic curves is the same as a nonzero homomorphism. An endomorphism of E is an isogeny from E to E.