Strata Hasse invariants, Hecke algebras and Galois representations

TL;DR

This paper develops a group-theoretical framework for Hasse invariants on Shimura varieties, leading to new insights into Galois representations, affine strata, and modular forms in the context of arithmetic geometry.

Contribution

It introduces a novel stack of zip flags and constructs generalized Hasse invariants, connecting geometric structures with automorphic and Galois representations.

Findings

Pseudo-representations attached to coherent cohomology.

Galois representations associated with automorphic forms.

All Ekedahl-Oort strata are proven to be affine.

Abstract

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^{\mu}$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^\mu$, fibered in flag varieties over $G$-Zip$^{\mu}$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four applications are obtained: (1) Pseudo-representations are attached to the coherent cohomology of Hodge-type Shimura varieties modulo a prime power. (2) Galois representations are associated to many automorphic representations with non-degenerate limit of discrete series archimedean component. (3) It is shown that all Ekedahl-Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine, thereby proving a conjecture of Oort. (4) Part of Serre's letter to Tate on mod $p$ modular forms is generalized to general Hodge-type Shimura varieties.