Xiaoyu Zhang: Distribution of reductions of abelian varieties modulo a prime

Number theory seminar: Xiaoyu Zhang (Universität Duisburg-Essen)

Distribution of reductions of abelian varieties modulo a prime

Given a family of abelian varieties over the rational numbers with good reductions at a fixed prime p, what can we say about the reductions modulo p of these abelian varieties? If they are all supersingular, do they run through all possible supersingular varieties? Such questions have interesting applications in Iwasawa theory and non-vanishing of L-values (for example the work of Cornut, Howard, Vatsal, etc). In this talk, I would like to discuss one such question for a family of abelian varieties parameterized by a Shimura variety and present some of my works in this direction and the application to Rankin-Selberg L-values.