Singularities in reductions of Shimura varieties

Singularities in reductions of Shimura varieties -Thomas Haines

Thomas Haines
University of Maryland
May 2, 2019

The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a fundamental
question to understand the singularities which arise in the reductions
modulo $p$ of integral models of Shimura varieties. For PEL type
Shimura varieties with parahoric level structure at $p$, this question
has been studied since the 1990's. Due to the recent construction of
Kisin and Pappas, it now makes sense to pursue this question for
abelian type Shimura varieties with parahoric level structure.
Recently He-Pappas-Rapoport gave a classification of the Shimura
varieties in this class which have either good or semistable
reduction. But what is the strongest statement we can make about the
nature of the singularities in general? For some time it has been
expected that the integral models are Cohen-Macaulay. This talk will
discuss recent work with Timo Richarz, in which we prove that, with
mild restrictions on $p$, all Pappas-Zhu parahoric local models, and
therefore all Kisin-Pappas Shimura varieties, are Cohen-Macaulay.