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 Av 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 Av 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, where m is an integer depending only on the endomorphism ring ${\mathrm End}(A_{\bar K})$. 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 $\mathbb Q$ generated by the Weil numbers of Av for most v.