On the Frobenius fields of abelian varieties over number fields

Let A be a non-CM simple abelian variety over a number field K.For a place v of K where A has good reduction, let F.A; v/ denote the Frobenius field generated by the corresponding Frobenius eigenvalues.If A has connected monodromy groups, we show that the set of places v such that F.A; v/ is isomorphic to a fixed number field has upper Dirichlet density zero.Moreover, assuming the GRH, we give a power saving upper bound for the number of such places.