Searcharxiv⌕ Search

arXiv subjects

Jean-Stefan Koskivirta

Publications and source records attributed to Jean-Stefan Koskivirta.

At least 19 recordsLinked to original sources
arXiv2026

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

For a connected, reductive group $G$ over a finite field endowed with a cocharacter $μ$, we define the zip cone of $(G,μ)$ 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.

math.NT↗
arXiv2026

Decidability of singularities in the Ekedahl--Oort stratification

For an abelian type Shimura variety and an odd prime $p$ of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an algorithm that detects codimension one singularities for arbitrary Ekedahl--Oort strata. When the Shimura datum is of split type, we relate the singularities of Ekedahl--Oort strata to a stack of $G$-zips over the complex numbers. We study the existence of generalized Hasse invariants on this stack.

math.NT↗
arXiv2025

Singularities in the Ekedahl--Oort stratification

We give conceptual and combinatorial criteria for the normality and Cohen--Macaulayness of unions of Ekedahl--Oort strata in the special fiber of abelian type Shimura varieties. For unions of two strata, one of the two having codimension one in the closure of the other, we determine exactly when their union is smooth. We provide explicit numerical criteria for the smoothness of any one-dimensional EO-stratum closure of Shimura varieties. For groups of type $\mathsf{B}_n$, we describe the smooth and normal loci of all EO-strata closures. We construct reduced strata Hasse invariants of explicit weights on EO-strata of codimension at most $n-1$, showing that their Zariski closures are local complete intersections. We also provide a closed form formula for their cycle classes.

math.NT↗
arXiv2024

The cone conjecture for vector-valued Siegel automorphic forms

We prove that the effective cone of automorphic vector bundles on the Siegel modular variety of rank $n$ in characteristic $p$ at a place of good reduction is encoded by the stack of $G$-zips of Pink--Wedhorn--Ziegler. Specifically, we show that the degree zero cohomology groups of automorphic vector bundles always vanish outside of the zip cone. This result is a special case of a general conjecture formulated by the Goldring and the author for all Hodge-type Shimura varieties of good reduction. In the case $n=3$, we give explicit conditions for the vanishing of the $0$-th cohomology group. Finally, in the course of the proof we define the notion of automorphic forms of trivial-type and study their properties.

math.NT↗
arXiv2024

The stack of $G$-zips is a Mori dream space

We first extend previous results of the author with T. Wedhorn and W. Goldring regarding the existence of $μ$-ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of $G$-zips of Pink--Wedhorn--Ziegler. Then, we define and study the Cox ring of the stack of $G$-zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of $G$-zips and study its properties. We prove that it is finitely generated in certain cases.

math.NT↗
arXiv2024

Partial Hasse invariants for Shimura varieties of Hodge-type

For a connected reductive group $G$ over a finite field, we define partial Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic forms which cut out a single codimension one stratum. We study their properties and show that such invariants admit a natural factorization through higher rank automorphic vector bundles. We define the socle of an automorphic vector bundle, and show that partial Hasse invariants lie in this socle.

math.AG↗
arXiv2023

Cohomology vanishing for Ekedahl--Oort strata on Hilbert--Blumenthal Shimura varieties

We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.

math.NT↗
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.

math.NT↗
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.

math.NT↗
arXiv2021

Automorphic vector bundles on the stack of $G$-zips

For a connected reductive group $G$ over a finite field, we study automorphic vector bundles on the stack of $G$-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski--Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of $G$-zips and a category of admissible modules with actions of a zero-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

math.NT↗
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.

math.NT↗
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.

math.NT↗
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.

math.AG↗
arXiv2017

Normalization of closed Ekedahl-Oort strata

We apply our theory of partial flag spaces developed in previous articles to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti-Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl-Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

math.AG↗
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.

math.NT↗