To an abelian variety over a number field one can associate an abelian variety to each prime ideal p of good reduction by reducing the variety$\mod p$. The geometry of these reductions need not resemble the geometry of the original abelian variety; for example, there are absolutely simple abelian varieties of dimension 2 whose reductions$\mod p$ always split as a product of elliptic curves. In this talk, we shall describe progress on a conjecture of Murty and Patankar which predicts exactly which absolutely simple abelian varieties have reductions modulo p that are also absolutely simple.