Strata Hasse invariants, Hecke algebras and Galois representations

By W. Goldring, Jean-Stefan Koskivirta. Inventiones Mathematicae, 2015

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We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $$\mathop {G\hbox {-}{} \mathtt{Zip}}\nolimits ^{\mu }$$G-Zipμ. 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 $$\mathop {G\hbox {-}{} \mathtt{ZipFlag}}\nolimits ^\mu $$G-ZipFlagμ, fibered in flag varieties over $$\mathop {G\hbox {-}{} \mathtt{Zip}}\nolimits ^{\mu }$$G-Zipμ. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25c3690794a195a7856aa8b6ccc03dcd319b8c1b