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Colorado State
Title:Split reductions of simple abelian varieties
Abstract:An irreducible, monic polynomial with integer coefficients is reducible modulo p for primes p in a set of positive density. One may pose an analogous problem for a simple abelian variety X over a number field; characterize those primes for which the reduction of X is isogenous to a product of abelian varieties of smaller dimension. This "splitting set" may have positive density or density zero, depending on the choice of X. Murty and Patankar conjecture that whether the splitting set has density zero depends only on the endomorphism ring of X. I'll explain a proof of part of their conjecture.