Uniqueness and functoriality of Igusa stacks

TL;DR

The paper axiomatizes Igusa stacks for arbitrary Shimura data by defining a v-stack $Igs(\mathsf{G}',\mathsf{X}')$ with a $\underline{\mathsf{G}(\mathbb{A}^{p,\infty})}$-action and a Cartesian diagram connecting the Shimura variety, the Grassmannian $\mathrm{Gr}_{G',\mathbb{Q}_p}$, and $\mathrm{Bun}_{G',\{\mu'^{-1}\}}$. It proves uniqueness up to unique isomorphism and functoriality under morphisms of Shimura data, and shows existence passes to subdata; for Hodge-type data, Igusa stacks exist on the good reduction locus. The paper also explains how Shimura-variety cohomology can be recovered from the Fargues–Scholze sheaf $\mathscr{F}=R\pi_{HT,*}\Lambda$ and outlines the construction of the Hodge–Tate period map via de Rham local systems and local shtukas, providing a unified framework across reduction types. It develops the global uniformization via the diagram with $\mathrm{BL}$ and the Beauville–Laszlo moduli, and lays groundwork for extending Igusa-stack methods to abelian-type data and bad reduction, connecting to Langlands–Rapoport-type perspectives through $p$-adic geometry. Overall, the work establishes a robust, functorial, axiomatic foundation for Igusa stacks and their role in understanding the cohomology and uniformization of Shimura varieties.

Abstract

We provide an axiomatic definition of an Igusa stack associated to an arbitrary Shimura datum. We then prove that Igusa stacks are unique and automatically functorial with respect to morphisms of Shimura data, assuming their existence. Using the same techniques, we also prove that the existence of the Igusa stack passes to a Shimura subdatum.