To an abelian variety over a number field one can associate another abelian variety to every prime ideal p of good reduction by reducing the variety modulo p. The geometry of these reductions need not resemble that of the original abelian variety; for example, a simple abelian variety might not have any reductions that are simple. In this talk, we shall describe progress on a conjecture of Murty and Patankar which classifies those simple abelian varieties that have reductions modulo p that are also simple.