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

TL;DR

This paper establishes a geometric relation between mod p reductions of quaternionic Shimura varieties with ramified primes and products of projective lines, extending the understanding of their stratifications.

Contribution

It introduces Pappas-Rapoport models at Iwahori level and proves isomorphisms of Goren-Oort strata to products of projective line bundles over auxiliary Shimura varieties.

Findings

Stratifications are isomorphic to products of $\

$ ext{P}^1$-bundles over auxiliary Shimura varieties.

Provides a geometric description of Goren-Oort strata at ramified primes.

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.