Thomas J. Haines
University of Maryland
Title
Local models for Shimura varieties
Abstract
The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will review the basic theory of local models for Iwahori level, concentrating on the cases related to general linear and symplectic groups. A second goal will be to explain what can be said about local models when the level structure is $\Gamma_1(p)$, which is slightly deeper than Iwahori level. For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order p, and then, if time, I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.