DATE: December 10, 2010 SPEAKER: Jeff Achter (Colorado State)

LOcation: PADELFORD C401.

TITLE: Split reductions of simple abelian varieties

Let X be an absolutely simple abelian variety over a number field. Then X may or may not have absolutely simple reduction almost everywhere; Murty and Patankar conjecture that this depends only on the endomorphism ring of X. I'll describe results towards this conjecture. In the case where X is an abelian surface with trivial endomorphism ring, I'll also describe support for a refined, quantitative conjecture.