Split reductions of simple abelian varieties
by
Jeff Achter
Colorado State University

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. I'll explain a proof of part of a conjecture of Murty and Patankar which explains this behavior.

Date received: April 17, 2008