Degree of definition of endomorphisms of an abelian variety

Given an abelian variety over a field of characteristic zero, we give an optimal explicit upper bound depending only on the dimension for the degree of the smallest extension of the base field over which all endomorphisms of the abelian variety are defined. For each dimension, the bound is achieved over the rationals by twisting a power of a CM elliptic curve. This complements a result of Guralnick and Kedlaya giving the exact value of the least common multiple of all these degrees. We also provide a similar statement for homomorphisms between two distinct abelian varieties. The proof relies on divisibility bounds obtained by Minkowski's method but, in some cases, we need more precise facts about finite linear groups, including a theorem of Feit whose proof has not been published I we therefore include one based on work by Collins on Jordan's theorem.