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.

# Singularities in reductions of Shimura varieties

Thomas Haines

University of Maryland

May 2, 2019