Explicit bounds for split reductions of simple abelian varieties

Let X/K be an absolutely simple abelian variety over a number field; we study whether the reductions Xp tend to be simple, too. We show that if End(X) is a definite quaternion algebra, then the reduction Xp is geometrically isogenous to the self-product of an absolutely simple abelian variety for p in a set of positive density, while if X is of Mumford type, then Xp is simple for almost all p. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.