Endomorphism fields of abelian varieties

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato–Tate group of the abelian variety, which had been proved for abelian surfaces by Fite–Kedlaya–Rotger–Sutherland. The proof uses Minkowski’s reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.