Abstract
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{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 mathop {Ghbox {-}{} mathtt{ZipFlag}}nolimits ^mu , fibered in flag varieties over mathop {Ghbox {-}{} mathtt{Zip}}nolimits ^{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.
Highlights
4.1.1 Rational theoryLet (G, X) be a Shimura datum [24, 2.1.1]
(2) Galois representations are associated to many automorphic representations with nondegenerate limit of discrete series Archimedean component
Starting with Ash [6], a number of torsion analogues of the Langlands correspondence have been proposed, where automorphic representations are replaced by systems of Hecke eigenvalues appearing in the cohomology (Betti or coherent) of a locally symmetric space with mod pn coefficients and the Galois representations (Gal) are replaced by mod pn-valued pseudorepresentations
Summary
By [62, Proposition 1.18], the space H 0([E\Gw] , V (χ )) is at most 1dimensional for all χ ∈ X ∗(L) and all w ∈ I W. The following assertions are equivalent: (a) There exists d ≥ 1 such that hdw,χ extends to E\Gw with non-vanishing locus [E\Gw]. By Theorem 2.2.1 (c), Property (b) is satisfied if and only if fw,Cλ extends to Sbtw with non-vanishing locus Sbtw. Let U ⊂ Y be a normal open subset and h ∈ L (U ) a non-vanishing section over U. Assume that the section f ∗(h) ∈ H 0( f −1(U ), f ∗L ) extends to X with non-vanishing locus f −1(U ). The parabolic subgroup z−1 Q contains B and has type J and Levi subgroup z−1 M It follows that α is not a root of z−1 M. Lemma 3.2.5 Let (G, μ) = (G Sp(W, ψ), μD) be a cocharacter datum of Siegel-type In [36], we have shown the analogue of Corollary 3.2.6 for any (G, μ) of Hodgetype (not necessarily attached to a Shimura datum), and for more general (G, μ) (those of maximal type)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
