Generalized μ-ordinary Hasse invariants

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.

show abstract

“…3 was proved in [62] for χ any L-ample character. The assumption that χ is orbitally p-close is superfluous in this case.…”

Section: We First Describe the Relation Between The Line Bundles Lmentioning

confidence: 94%

“…We thank our respective collaborators Marc-Hubert Nicole and Torsten Wedhorn: The joint projects [39] and [62] were invaluable in leading us to the results of the current paper. We are grateful to David Geraghty for the many important ideas that he contributed during the initial part of this project.…”

Section: Acknowledgements 45mentioning

confidence: 98%

Strata Hasse invariants, Hecke algebras and Galois representations

Goldring

Koskivirta

2019

Invent. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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].…”

Section: Hasse Invariantsmentioning

confidence: 99%

A Quotient of the Lubin–tate Tower

Ludwig

2017

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

In this article we show that the quotient M ∞ /B(Q p ) of the Lubin-Tate space at infinite level M ∞ by the Borel subgroup of upper triangular matrices B(Q p ) ⊂ GL 2 (Q p ) exists as a perfectoid space. As an application we show that Scholze's functoris concentrated in degree one whenever π is an irreducible principal series representation or a twist of the Steinberg representation of GL 2 (Q p ).

show abstract

“…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 [3,10,13,25]; moreover the works [4,9] 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 [3].…”

Section: Hasse Invariantsmentioning

confidence: 99%

A quotient of the Lubin–Tate tower II

Ludwig

2020

Math. Ann.

View full text Add to dashboard Cite

In this article we construct the quotient $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) of the infinite-level Lubin–Tate space $$\mathcal {M}_\mathbf {1}$$ M 1 by the parabolic subgroup $$P(K) \subset \mathrm {GL} _n(K)$$ P ( K ) ⊂ GL n ( K ) of block form $$(n-1,1)$$ ( n - 1 , 1 ) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and $$K/{\mathbb {Q}} _p$$ K / Q p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) when $$n=2$$ n = 2 , and shows that $$\mathcal {M}_\mathbf {1}/Q(K)$$ M 1 / Q ( K ) is not perfectoid for maximal parabolics Q not conjugate to P.

show abstract

Generalized μ-ordinary Hasse invariants

Cited by 17 publications

References 8 publications

Strata Hasse invariants, Hecke algebras and Galois representations

Strata Hasse invariants, Hecke algebras and Galois representations

A Quotient of the Lubin–tate Tower

A quotient of the Lubin–Tate tower II

Contact Info

Product

Resources

About