Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A mod $p$ Geometric Jacquet-Langlands Relation for Quaternionic Shimura Varieties at Ramified Primes

Gabriel Micolet

TL;DR

The paper studies p-adic integral models for Quaternionic Shimura varieties attached to a quaternion algebra B/F when p ramifies in F but B splits at places over p. It develops Pappas–Rapoport splitting models, a ramified Goren–Oort stratification via partial Hasse invariants, and Iwahori level structures, then extends Raynaud/crystalline techniques to define partial Raynaud subgroups and essential Frobenius isogenies. The central achievement is a geometric Jacquet–Langlands relation mod p: each smooth Goren–Oort stratum upstairs at Iwahori level is canonically isomorphic to a product of $\mathbb{P}^1$-bundles over auxiliary Quaternionic Shimura varieties, obtained by splicing Dieudonné modules and filtering subobjects. This splicing/splitting framework yields explicit moduli-theoretic descriptions that generalize previous unramified and Hilbert settings to ramified primes, enabling a modular interpretation of the GO strata and a robust link between unitary, Hilbert, and quaternionic models with applications to mod p automorphic phenomena.

Abstract

Let $F$ be a totally real field, $p$ a prime that we allow to ramify in $F$, and $B$ a quaternion algebra over $F$ which is split at places over $p$. We consider a smooth $p$-adic integral model, the Pappas-Rapoport model, of the Quaternionic Shimura variety attached to $B$ with prime-to-$p$ level, and the Goren-Oort stratification of its characteristic $p$ fiber. Furthermore, we also introduce Pappas-Rapoport models at Iwahori level $p$ along with a stratification of their characteristic $p$ fiber. We prove that these strata are isomorphic to products of $\mathbb{P}^1$-bundles over auxiliary Quaternionic Shimura varieties, from which we deduce the corresponding description of the Goren-Oort strata.

A mod $p$ Geometric Jacquet-Langlands Relation for Quaternionic Shimura Varieties at Ramified Primes

TL;DR

The paper studies p-adic integral models for Quaternionic Shimura varieties attached to a quaternion algebra B/F when p ramifies in F but B splits at places over p. It develops Pappas–Rapoport splitting models, a ramified Goren–Oort stratification via partial Hasse invariants, and Iwahori level structures, then extends Raynaud/crystalline techniques to define partial Raynaud subgroups and essential Frobenius isogenies. The central achievement is a geometric Jacquet–Langlands relation mod p: each smooth Goren–Oort stratum upstairs at Iwahori level is canonically isomorphic to a product of -bundles over auxiliary Quaternionic Shimura varieties, obtained by splicing Dieudonné modules and filtering subobjects. This splicing/splitting framework yields explicit moduli-theoretic descriptions that generalize previous unramified and Hilbert settings to ramified primes, enabling a modular interpretation of the GO strata and a robust link between unitary, Hilbert, and quaternionic models with applications to mod p automorphic phenomena.

Abstract

Let be a totally real field, a prime that we allow to ramify in , and a quaternion algebra over which is split at places over . We consider a smooth -adic integral model, the Pappas-Rapoport model, of the Quaternionic Shimura variety attached to with prime-to- level, and the Goren-Oort stratification of its characteristic fiber. Furthermore, we also introduce Pappas-Rapoport models at Iwahori level along with a stratification of their characteristic fiber. We prove that these strata are isomorphic to products of -bundles over auxiliary Quaternionic Shimura varieties, from which we deduce the corresponding description of the Goren-Oort strata.
Paper Structure (79 sections, 57 theorems, 361 equations)

This paper contains 79 sections, 57 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. Modular curves
  3. Hilbert modular varieties at unramified primes
  4. Pappas-Rapoport models of Hilbert modular varieties
  5. Overview of the thesis
  6. Tame level Shimura varieties
  7. The Goren-Oort stratification
  8. Iwahori level models
  9. Geometric Jacquet Langlands relations mod $p$
  10. Overview and advice for the Reader
  11. Acknowledgements
  12. Tame Level Shimura varieties
  13. Pappas-Rapoport models of Hilbert Modular varieties
  14. Notation
  15. The models
  16. ...and 64 more sections

Key Result

Theorem 1.1

There is a bijection of sets equivariant under Hecke correspondences, where $B_{p,\infty}$ denotes the unique Quaternion algebra over $\mathbb{Q}$ ramified at $p$ and infinity and $O_1(N) \subset B_{p,\infty}^\times(\mathbb{A}^{(p)}_f)$ is the image of $\Gamma_1(N)$ under the isomorphism $B_{p,\infty}^\times(\mathbb{A}^{(p)}_f)

Theorems & Definitions (105)

  • Theorem 1.1: Serre
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 95 more