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The singularities in the reduction modulo $p$ of the modular curve $Y_0(p)$ are visualized by the famous picture of two curvesÌýmeeting transversally at the supersingular points. It is a fundamentalÌýquestion to understand the singularities which arise in the reductionsÌýmodulo $p$ of integral models of Shimura varieties. For PEL typeÌýShimura varieties with parahoric level structure at $p$, this questionÌýhas been studied since the 1990's. Due to the recentÌý construction ofÌýKisin and Pappas, it now makes sense to pursue this question forÌýabelian type Shimura varieties with parahoric level structure. Recently He-Pappas-Rapoport gave a classification of the ShimuraÌývarieties in this class which have either good or semistableÌýreduction. But what is the strongest statement we can make about theÌýnature of the singularities in general?Ìý For some time it has beenÌýexpected that the integral models are Cohen-Macaulay.

This talk willÌýdiscuss recent work with Timo Richarz, in which we prove that, withÌýmild restrictions on $p$, all Pappas-Zhu parahoric local models, andÌýtherefore all Kisin-Pappas Shimura varieties, are Cohen-Macaulay.