The Splitting of Reductions of an Abelian Variety

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A(v) of A modulo v splits up into isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing the field K, we will show that A(v) is isogenous to the mth power of an absolutely simple abelian variety for all places v of K away from a set of density 0, wheremis an integer depending only on the endomorphism ring End(A((K) over bar)). This proves many cases, and supplies justification, of a conjecture of Murty and Patankar. Under the same assumptions, we will also describe the Galois extension of Q generated by the Weil numbers of A(v) for most v.