Endomorphism rings of Abelian varieties and their representations

These are notes of two talks with the aim of giving some basic properties of the endomorphism ring of an Abelian variety A and its representations on certain linear objects associated to A. The results can be found in § 5.1 of Shimura’s book [1], but presented in a completely different way. For completeness, we state some definitions. An Abelian variety over a field k is a proper, smooth, connected group variety over k. A basic result from the theory of Abelian varieties is that every Abelian variety is commutative (and projective, but we will not use this.) A homomorphism between Abelian varieties A and B is a morphism A → B of varieties over k that is compatible with the group structure. The set Hom(A,B) of all homomorphisms from A to B is an Abelian group, and the group EndA of all endomorphisms of A is a ring. An isogeny between Abelian varieties is a surjective homomorphism with finite kernel. An Abelian variety A is simple if it has exactly two Abelian subvarieties (namely 0 and A).