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.