The PI will continue his work on the arithmetic of Shimura varieties and applications. One goal is to apply the PI's recent results on the mod p points on these varieties to compute their zeta functions. Another aim is to extend some of these results to cases of bad reduction. One possible application is a result about the structure of de Rham-Tate cycles on abelian varieties.
Shimura varieties are a class of geometric objects which have played a very important role in many advances in number theory during the past 50 years. They can be thought of as parameter (or moduli) spaces for abelian varieties. The project aims to study the number theoretic properties of Shimura varieties, and their applications to the arithmetic of abelian varieties.
