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$^{\mu}$. 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 $G$-ZipFlag$^\mu$, fibered in flag varieties over $G$-Zip$^{\mu}$. 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.

Insightful Overview of "Strata Hasse Invariants, Hecke Algebras and Galois Representations"

The paper "Strata Hasse invariants, Hecke algebras, and Galois representations" by Wushi Goldring and Jean-Stefan Koskivirta presents a comprehensive study of the interactions between certain structures in algebraic geometry and number theory. The primary focus of the paper is to construct group-theoretical generalizations of the Hasse invariant on strata closures of algebraic stacks related to Shimura varieties, extending concepts that are pivotal in the study of automorphic forms and Galois representations.

Group-Theoretical Hasse Invariants

The authors start by constructing Hasse invariants on strata closures of the stacks $^{\mu}$ using group-theoretical methods. These invariants are notable for their ability to generalize classical notions and are essential for understanding the stratification of Shimura varieties. They are derived by considering the Ekedahl-Oort stratification, a vital aspect of the structure of Shimura varieties mod $p$, which sorts said varieties according to the associated $p$-divisible groups.

Four Applications

1. Pseudo-representations Modulo Prime Power: The paper attaches pseudo-representations to coherent cohomology of Hodge-type Shimura varieties modulo a prime power. This association is a substantial step in understanding the representation theory related to these varieties, particularly in the context of torsion.
2. Galois Representations: Goldring and Koskivirta associate Galois representations to automorphic representations with non-degenerate limits of discrete series components. This application extends prior work, offering insights into how certain automorphic forms relate to Galois representations.
3. Affineness of Hodge-Type Shimura Variety Strata: The authors prove that all Ekedahl-Oort strata in the minimal compactification of a Hodge-type Shimura variety are affine. This settles conjectures posed by Oort related to the affineness of strata, providing rigorous geometric insights into their structure.
4. Generalization of Serre's Letter to Tate: The final application generalizes components of Serre's famous letter to Tate, extending its results to Hodge-type Shimura varieties. This generalization is central to understanding modular forms in characteristic $p$.

Implications and Future Directions

The results presented in this paper have significant implications for the theoretical framework connecting automorphic algebraicity and $G$-Zip geometricity. By establishing novel relations and results, Goldring and Koskivirta’s work opens new pathways for research in both algebraic geometry and number theory, particularly in the study of algebraic cycles, automorphic forms, and representations.

Moreover, the techniques developed for constructing Hasse invariants and analyzing their implications in the presence of various strata within Shimura varieties provide a new lens through which researchers can explore modular curves and other related algebraic structures.

Conclusion

This paper represents a robust and methodical approach to expanding our understanding of Shimura varieties' structure and their connection to broader concepts in number theory. By extending classical notions and constructing novel invariants, the authors have laid essential groundwork for future research. As the field of algebraic geometry continues to evolve, papers like this set a valuable precedent for the depth of analysis and sophistication required to tackle complex mathematical challenges.