Strata Hasse invariants, Hecke algebras and Galois representations

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 Hodgetype 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.

Minimal weights of Hilbert modular forms in characteristic

“…with torsion-free cokernel (by construction). We already saw in the proof of Theorem 6.1.1 that the map M k,0 (U ; E) → M m+2,0 (U ′ ; O) ⊗ O E is compatible with the operators T v and S v for v ∤ pn, and it follows that the same holds for the composite on the top row of (34). We claim that this composite is also compatible with the operators T v for v|p.…”

“…This has been proved for paritious weights (k, l), independently by Emerton-Reduzzi-Xiao [24] and Goldring-Koskivirta [34]; in fact their methods yield the result under a weaker parity condition. The contribution here is to remove the parity hypothesis altogether, and the new ingredient is to use congruences to forms of level divisible by p. For this we will need to work with the integral models for Hilbert modular varieties with level structure U 1 (p) at p studied by Pappas in [48].…”

A Quotient of the Lubin–tate Tower

“…The topic of generalized Hasse invariants has received a lot of attention recently. In the case of µ-ordinary Hasse invariants we mention the works [GN17, KW18,Her16,BH17]; moreover the works [GK16,Box15] construct generalized Hasse invariants on all Ekedahl-Oort strata (in the cases when they apply). In particular, µ-ordinary Hasse invariants have been defined in large generality (including the cases needed here) by Bijakowski and Hernandez [BH17].…”

“…We need to show that ̟ : G x → G x factors through F r q : G x → G Remark 2.2.4. The proof above works to give 'strata' Hasse invariants cutting out the Ekedahl-Oort strata, in the sense of [Box15,GK16]. These strata Hasse invariants were already defined by Ito [Ito,Ito06].…”

Groupesp-divisibles avec condition de Pappas-Rapoport et invariants de Hasse

“…On peut aussi associer à G un invariant de Hasse µ-ordinaire, qui est une section sur S d'un fibré en droites, le déterminant du faisceau conormal de G (à une certaine puissance explicite) ; voir [Her16] pour la construction à l'aide de la cohomologie cristalline de G, ou [KW14] pour une construction utilisant les G-Zip. Ces derniers invariants ont de nombreuses applications en dehors de la géométrie des variétés de Shimura ([DS74] dans le cas de la courbe modulaire, puis [Box15,GK15] dans un cadre plus général les utilisent pour construire des représentations galoisiennes par interpolation).…”