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

TL;DR

This paper describes the structure of certain p-adic Shimura varieties using fiber products of stacks, extending known results and conjectures in p-adic geometry and Shimura variety theory.

Contribution

It constructs a fiber product description of the good reduction locus of PEL-type Shimura varieties as a stack, extending to minimal compactifications and integral models.

Findings

Describes the good reduction locus as a fiber product with an Igusa stack.

Constructs a minimal compactification of the Igusa stack.

Recovers the integral model of the Shimura variety via fiber product with shtukas.

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.