Igusa stacks for certain abelian-type Shimura varieties
Fabian Schnelle
TL;DR
The paper refines Scholze’s fiber product program for Shimura varieties with infinite $p$-level by constructing Igusa stacks for abelian-type cases arising from PEL data of type $(A ext{ even})$ or $(C)$ unramified at $p$. It introduces a two-step framework: first build hybrid spaces and a hybrid Igusa stack at infinite level, then quotient by a polarization action to obtain the abelian-type Igusa stack; the quotient is shown to be pro-étale after passing to uniformly good level subgroups. Key technical ingredients include perfectoid/diamond/v-stack technology, the Hodge–Tate period map, the Beauville–Laszlo gluing for $G$-bundles on the Fargues–Fontaine curve, and prismatic Dieudonné theory for $p$-divisible groups. The main result is a Cartesian fiber-product diagram for the abelian-type Shimura varieties, mirroring the PEL-derived structure and enabling a detailed comparison of $p$-adic geometry with Hecke actions. These constructions provide a concrete, functorial model of Igusa stacks in the abelian-type setting and pave the way for further cohomological and geometric applications in $p$-adic Hodge theory and Shimura varieties.
Abstract
We construct Igusa stacks for the good reduction locus of a class of abelian-type Shimura varieties that can be defined in terms of a PEL datum, under the assumption that it is of type (A even) or (C) and unramified at a prime p.