Basic loci of Shimura varieties

Basic loci of Shimura varieties

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Xuhua He , IAS
IAS Room S-101

Please note location (IAS) and special start time:  4:15 pm.   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.