$F$-zips with additional structure on splitting models of Shimura varieties

TL;DR

This paper constructs universal G-zips on splitting models of Shimura varieties, analyzes their stratification, and extends these structures to compactifications, with applications to Galois representations in the ramified setting.

Contribution

It introduces new universal G-zips on splitting models, studies their stratification, and extends constructions to compactifications, linking to Galois representations.

Findings

New universal G-zips constructed on splitting models.

Detailed analysis of Ekedahl-Oort stratification.

Application to Galois representations from torsion classes.

Abstract

We construct universal $G$-zips on good reductions of the Pappas-Rapoport splitting models for PEL-type Shimura varieties. We study the induced Ekedahl-Oort stratification, which sheds new light on the mod $p$ geometry of splitting models. Building on the work of Lan on arithmetic compactifications of splitting models, we further extend these constructions to smooth toroidal compactifications. Combined with the work of Goldring-Koskivirta on group theoretical Hasse invariants, we get an application to Galois representations associated to torsion classes in coherent cohomology in the ramified setting.