Fields of definition for homomorphisms of abelian varieties

AbstractWe give results on when homomorphisms between abelian varieties are or are not defined over fields obtained from division points on the varieties. For example, if A and B are abelian varieties defined over a field F, of dimensions d and e, respectively, and L is the intersection of the fields F(AN, BN) for all integers N prime to the characteristic of F and greater than 2, then every element of Hom(A, B) is defined over L,LF is unramified at the discrete places of good reduction for A × B, and [L : F] divides H(d,e), where H(d,e) is a number given by an explicit formula and is less than 4(9d)2d(9e)2e