Basic loci of Shimura varieties

Joint IAS/Princeton University Number Theory Seminar
Topic:Basic loci of Shimura varieties
Speaker:Xuhua He
Affiliation:University of Maryland; von Neumann Fellow, School of Mathematics
Date:Thursday, April 6
Time/Room:4:15pm - 5:15pm/S-101
Video Link:https://video.ias.edu/puias/2017/0406-XuhuaHe

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves. In certain cases, the basic locus admits a simple description as a union of classical Deligne-Lusztig varieties. The precise description in these case has proved to be useful for several purposes: to compute intersection numbers of special cycles and to prove the Tate conjecture for certain Shimura varieties. We will describe a group-theoretic approach to understand this phenomenon. We will show that this phenomenon is closely related to the Hodge-Newton decomposition, and many other nice properties on the Shimura varieties. This talk is based on the joint work with Ulrich Gortz and Sian Nie.