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A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing

Fred Diamond, Payman Kassaei, Shu Sasaki

TL;DR

This work constructs a geometric Jacquet--Langlands relation for Hilbert modular varieties with Iwahori level at $p$, showing that irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties and enabling a direct splicing of Dieudonné modules. By combining this with a Serre-type filtration via a Kodaira--Spencer/Frobenius framework and a detailed analysis of degeneracy fibres through crystalline Dieudonné theory, the authors relate mod $p$ Hilbert modular forms to quaternionic automorphic forms and establish a cohomological vanishing result essential for attaching Galois representations in characteristic $p$. Their approach avoids Frobenius-factor detours, using splicing to achieve a more natural interplay between mod $p$ geometries of different reductive groups. The results generalize Serre's weight phenomenon to Hilbert modular contexts and illuminate how automorphic-bundle data transform under level-raising and component-wise identifications, with significant implications for the construction of mod $p$ Galois representations. Overall, the paper provides new geometric bridges between Hilbert, unitary, and quaternionic Shimura varieties and develops tools (dicing, splicing) that could extend to higher-rank settings and broader groups.

Abstract

We consider Hilbert modular varieties in characteristic p with Iwahori level at p and construct a geometric Jacquet-Langlands relation showing that the irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties of level prime to p. We use this to establish a relation between mod p Hilbert and quaternionic modular forms that reflects the representation theory of GL_2 in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibres of the degeneracy map to level prime to p and prove a cohomological vanishing result that is used to associate Galois representations to mod p Hilbert modular forms.

A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing

TL;DR

This work constructs a geometric Jacquet--Langlands relation for Hilbert modular varieties with Iwahori level at , showing that irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties and enabling a direct splicing of Dieudonné modules. By combining this with a Serre-type filtration via a Kodaira--Spencer/Frobenius framework and a detailed analysis of degeneracy fibres through crystalline Dieudonné theory, the authors relate mod Hilbert modular forms to quaternionic automorphic forms and establish a cohomological vanishing result essential for attaching Galois representations in characteristic . Their approach avoids Frobenius-factor detours, using splicing to achieve a more natural interplay between mod geometries of different reductive groups. The results generalize Serre's weight phenomenon to Hilbert modular contexts and illuminate how automorphic-bundle data transform under level-raising and component-wise identifications, with significant implications for the construction of mod Galois representations. Overall, the paper provides new geometric bridges between Hilbert, unitary, and quaternionic Shimura varieties and develops tools (dicing, splicing) that could extend to higher-rank settings and broader groups.

Abstract

We consider Hilbert modular varieties in characteristic p with Iwahori level at p and construct a geometric Jacquet-Langlands relation showing that the irreducible components are isomorphic to products of projective bundles over quaternionic Shimura varieties of level prime to p. We use this to establish a relation between mod p Hilbert and quaternionic modular forms that reflects the representation theory of GL_2 in characteristic p and generalizes a result of Serre for classical modular forms. Finally we study the fibres of the degeneracy map to level prime to p and prove a cohomological vanishing result that is used to associate Galois representations to mod p Hilbert modular forms.

Paper Structure

This paper contains 89 sections, 31 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Geometric Jacquet--Langlands
  4. The Serre filtration
  5. Degeneracy fibres
  6. Questions
  7. Acknowledgments
  8. Shimura varieties
  9. Hilbert modular varieties
  10. The Shimura datum
  11. The moduli problem
  12. Descent and Hecke action
  13. Unitary Shimura varieties
  14. The Shimura data
  15. The moduli problem
  16. ...and 74 more sections

Key Result

Theorem A

For each $J \subset \overline{\Theta}_p$ and sufficiently small open compact subgroup $U$ of $\mathop{\mathrm{GL}}\nolimits_2(\mathbf{A}_{F,\mathbf{f}})$ containing $\mathop{\mathrm{GL}}\nolimits_2(O_{F,p})$, there is a Hecke-equivariant isomorphism where the product is a fibre product over ${\overline{Y}}_\Sigma$, and $\Sigma = \Sigma_J \subset \Theta_\infty$ corresponds under the identification

Theorems & Definitions (38)

  • Theorem A
  • Theorem B
  • Corollary C
  • Theorem D
  • Theorem E
  • Lemma 2.2.1
  • Lemma 2.3.1
  • Lemma 2.3.2
  • Remark 3.1.1
  • Lemma 4.1.1
  • ...and 28 more