Splitting of Abelian Varieties

We study a new local–global problem in the context of Abelian varieties: Given an absolutely simple Abelian variety over a number field K, find a necessary and sufficient condition for the existence of a place v of K (or infinitely many places, or a set of places of positive density) such that A remains absolutely simple modulo v. It is well known that for absolutely simple Abelian surfaces with multiplication by an indefinite quaternion algebra, A modulo v, denoted by Av, is always isogenous to the square of some elliptic curve. For absolutely simple Abelian varieties of complex multiplication type, we show that Av stays simple for a set of places of density 1. On the other hand, for absolutely simple Abelian varieties associated by Shimura to cusp forms of weight 2, if Av splits at a set of places of positive density, then the absolute endomorphism algebra is noncommutative. Based on these results, we then formulate a conjecture: An absolutely simple Abelian variety defined over a number field splits at a set of places of positive density if and only if its absolute endomorphism algebra is noncommutative.