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Wushi Goldring

Publications and source records attributed to Wushi Goldring.

13 recordsLinked to original sources
arXiv2024

An Ogus Principle for Zip period maps: The Hasse invariant's vanishing order via `Frobenius and the Hodge filtration'

This paper generalizes a result of Ogus that, under certain technical conditions, the vanishing order of the Hasse invariant of a family $Y/X$ of $n$-dimensional Calabi-Yau varieties in characteristic $p$ at a point $x$ of $X$ equals the "conjugate line position" of $H^n_{\text{dR}}(Y/X)$ at $x$, i.e. the largest $i$ such that the line of the conjugate filtration is contained in $\text{Fil}^i$ of the Hodge filtration. For every triple $(G,μ,r)$ consisting of a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $μ\in X_*(G)$ and an $\mathbf{F}_p$-representation $r$ of $G$, we state a generalized Ogus Principle. If $ζ:X \to \text{$G$-$\mathtt{Zip}$}^μ$ is a smooth morphism (=`Zip period map'), then the group theoretic Ogus Principle implies an Ogus Principle on $X$. We deduce an Ogus Principle for several Hodge and abelian-type Shimura varieties and the moduli space of K3 surfaces.

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arXiv2024

Hodge-Chern classes and strata-effectivity in tautological rings

Given a connected, reductive $\mathbf{F}_p$-group $G$, a cocharacter $μ\in X_*(G)$ and a smooth zip period map $ζ:X \to \mathop{\text{$G$-{\tt Zip}}}\nolimits^μ$, we study which classes in the Wedhorn-Ziegler tautological rings $T^*(X), T^*(Y)$ of $X$ and its flag space $Y \to G-ZipFlag^μ$ are \textit{strata-effective}, meaning that they are non-negative rational linear combinations of pullbacks of classes of zip (flag) strata closures. Two special cases are: (1) When $X=G\text{-Zip}^μ$ and the tautological rings $\T^*(X)=\text{CH}_{\mathbf{Q}}(G-Zip^μ)$, $T^*(Y)=\text{CH}_{\mathbf{Q}}(G-ZipFlag^μ)$ are the entire Chow ring, and (2) When $X$ is the special fiber of an integral canonical model of a Hodge-type Shimura variety -- in this case the strata are also known as Ekedahl-Oort strata. We focus on the strata-effectivity of three types of classes: (a) Effective tautological classes, (b) Chern classes of Griffiths-Hodge bundles and (c) Generically $w$-ordinary curves. We connect the question of strata-effectivity in (a) to the global section `Cone Conjecture' of Goldring-Koskivirta. For every representation $r$ of $G$, we conjecture that the Chern classes of the Griffiths-Hodge bundle associated to $(G, μ,r)$ are all strata-effective. This provides a vast generalization of a result of Ekedahl-van der Geer that the Chern classes of the Hodge vector bundle on the moduli space of principally polarized abelian varieties $\Acal_{g,\mathbf{F}_p}$ in characteristic $p$ are represented by the closures of $p$-rank strata. We prove several instances of our conjecture

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arXiv2022

Divisibility of mod $p$ automorphic forms and the cone conjecture for certain Shimura varieties of Hodge-type

For several Hodge-type Shimura varieties of good reduction in characteristic $p$, we show that the cone of weights of automorphic forms is encoded by the stack of $G$-zips of Pink-Wedhorn-Ziegler. This establishes several instances of a general conjecture formulated in previous papers by the authors. Furthermore, we prove in these cases that any mod $p$ automorphic form whose weight lies in a specific region of the weight space is divisible by a partial Hasse invariant. This generalizes to other Shimura varieties previous results of Diamond--Kassaei on Hilbert modular forms.

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arXiv2022

Griffiths-Schmid conditions for automorphic forms via characteristic $p$

We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.

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arXiv2022

Weights of mod $p$ automorphic forms and partial Hasse invariants

For a connected, reductive group $G$ over a finite field endowed with a cocharacter $\mu$, we define the zip cone of $(G,\mu)$ as the cone of all possible weights of mod $p$ automorphic forms on the stack of $G$-zips. This cone is conjectured to coincide with the cone of weights of characteristic $p$ automorphic forms for Hodge-type Shimura varieties of good reduction. We prove in full generality that the cone of weights of characteristic $0$ automorphic forms is contained in the zip cone, which gives further evidence to this conjecture. Furthermore, we determine exactly when the zip cone is generated by the weights of partial Hasse invariants, which is a group-theoretical generalization of a result of Diamond--Kassaei and Goldring--Koskivirta.

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arXiv2020

Quasi-constant fundamental weights in terms of Levi Weyl groups

In joint work with J.-S. Koskivirta, we had previously introduced the notion of "quasi-constant" character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions "minuscule" and "co-minuscule". In this note we describe a characterization of quasi-constant fundamental weights in terms of the Weyl group of a maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of Weyl groups of maximal sub-root systems.

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arXiv2019

The Griffiths bundle is generated by groups

First the Griffiths line bundle of a $\mathbf Q$-VHS $\mathscr V$ is generalized to a Griffiths character ${\rm grif}(\mathbf G, μ,r)$ associated to any triple $(\mathbf G, μ, r)$, where $\mathbf G$ is a connected reductive group over an arbitrary field $F$, $μ\in X_*(\mathbf G)$ is a cocharacter (over $\overline{F}$) and $r:\mathbf G \to GL(V)$ is an $F$-representation; the classical bundle studied by Griffiths is recovered by taking $F=\mathbf Q$, $\mathbf G$ the Mumford-Tate group of $\mathscr V$, $r:\mathbf G \to GL(V)$ the tautological representation afforded by a very general fiber and pulling back along the period map the line bundle associated to ${\rm grif}(\mathbf G, μ, r)$. The more general setting also gives rise to the Griffiths bundle in the analogous situation in characteristic $p$ given by a scheme mapping to a stack of $\mathbf G$-Zips. When $\mathbf G$ is $F$-simple, we show that, up to positive multiples, the Griffiths character ${\rm grif}(\mathbf G,μ,r)$ (and thus also the Griffiths line bundle) is essentially independent of $r$ with central kernel, and up to some identifications is given explicitly by $-μ$. As an application, we show that the Griffiths line bundle of a projective $\mathbf G{\rm -Zip}^μ$-scheme is nef.

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arXiv2019

Strata Hasse invariants, Hecke algebras and Galois representations

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.

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arXiv2018

Quasi-constant characters: Motivation, classification and applications

In our previous paper "Strata Hasse invariants, Hecke algebras and Galois representations", initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of {\em minuscule character} which we termed {\em quasi-constant}. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if $μ$ is a quasi-constant cocharacter of an ${\mathbf F}_p$-group $G$, then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack $G\mbox{-Zip}^μ$, without any restrictions on $p$, even if the pair $(G, μ)$ is not of Hodge type and even if $μ$ is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc cit.

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arXiv2017

Stratifications of flag spaces and functoriality

We define quotient stacks of "zip flags". They form towers above the stack of $G$-zips introduced by Moonen, Pink, Wedhorn and Ziegler. We define a stratification on the stack of zip flags, and prove that it is principally pure, under a certain assumption on $p$. The fiber product with a Shimura variety of Hodge-type is a generalization of flag spaces considered by Ekedahl-Van der Geer. For large $p$, we prove that all strata are affine. We prove a theorem on discreteness of fibers for finite morphisms between stacks of $G$-zips. This allows us to prove that the zip stratification is principally pure for all primes $p$, for zip data of Hodge-type. This provides a second proof, entirely in characteristic $p$, of the existence of generalized Hasse invariants for Ekedahl-Oort strata in the good reduction of Shimura varieties of Hodge-type.

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arXiv2017

Automorphic vector bundles with global sections on $G$-${\tt Zip}^{\mathcal Z}$-schemes

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of $G$-zips of connected-Hodge-type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type $A_1^n$, $C_2$ and $\mathbf F_p$-split groups of type $A_2$ (this includes all Hilbert-Blumenthal varieties and should also apply to Siegel modular threefolds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected-Hodge-type.

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arXiv2014

A μ-ordinary Hasse invariant

We construct a generalization of the Hasse invariant for certain unitary Shimura varieties of PEL type whose vanishing locus is the complement of the so-called μ-ordinary locus. We show that the μ-ordinary locus of those varieties is affine. As an application, we strengthen a special case of a theorem of one of us (W.G.) on the association of Galois representations to automorphic representations of unitary groups whose archimedean component is a holomorphic limit of discrete series.

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