Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence

TL;DR

This paper explores the cohomological construction of p-adic Jacquet-Langlands correspondence for Shimura curves associated with indefinite quaternion algebras over totally real fields, introducing new techniques and level lowering principles.

Contribution

It provides a novel cohomological approach to p-adic Jacquet-Langlands functoriality and establishes an analogue of Mazur's level lowering for these automorphic forms.

Findings

Constructed a cohomological p-adic Jacquet-Langlands correspondence.

Established a level lowering principle for p-adic automorphic forms.

Provided explicit examples within quaternion algebra contexts.

Abstract

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur's `level lowering' principle.