Strata Hasse invariants, Hecke algebras and Galois representations

2015/07/17 by Goldring, Wushi, Koskivirta, Jean-Stefan · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1507.05032

Abstract

We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks G-Zip^μ. 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^μ, fibered in flag varieties over G-Zip^μ. 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.

Cited by

• Irreducibility of Newton strata in Picard modular surfaces and split local Galois representations
• Singularities in the Ekedahl--Oort stratification

Related

• Stratifications of flag spaces and functoriality
• The stack of G-zips is a Mori dream space
• F-zips with additional structure on splitting models of Shimura varieties
• An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'
• A μ-ordinary Hasse invariant