A PEL-type Igusa Stack and the $p$-adic Geometry of Shimura Varieties

Abstract

Let $(G,X)$ be a PEL-Shimura datum of type AC in Kottwitz's classification. Assume $G_{\mathbf{Q}_p}$ is unramified. We show that the good reduction locus of the infinite $p$-level Shimura variety attached to this datum, considered as a diamond, can be described as the fiber product of a certain v-stack (which we call ``Igusa stack") with a Schubert cell of the corresponding $B_{dR}^+$-affine Grassmannian, over the stack of $G_{\mathbf{Q}_p}$-torsors on the Fargues-Fontaine curve. We also construct a minimal compactification of the Igusa stack and show that this fiber product structure extends to the minimal compactification of the Shimura variety. When the Schubert cell of the affine Grassmannian is replaced by a bounded substack of $\mathcal{G}$-shtukas, where $\mathcal{G}$ is a reductive model of $G_{\mathbf{Q}_p}$ over $\mathbf{Z}_p$, we show that this fiber product recovers the integral model of the Shimura variety. This result on integral models, if specialized to a Newton polygon stratum, recovers the fiber product formula of Mantovan. Similar fiber product structures are conjectured by Scholze to exist on general Shimura varieties.Comment: Comments welcome