The Jacquet-Langlands correspondence for overconvergent Hilbert modular forms

TL;DR

This paper extends the classical Jacquet-Langlands correspondence to overconvergent Hilbert modular forms over totally real fields, establishing isomorphisms between associated eigenvarieties.

Contribution

It generalizes Chenevier's results to totally real fields and constructs isomorphisms between eigenvarieties for Hilbert modular forms and quaternion algebra forms.

Findings

Established isomorphisms between eigenvarieties for Hilbert modular forms and quaternion algebra forms.

Extended Jacquet-Langlands correspondence to overconvergent forms over totally real fields.

Provided a framework for interpolating classical correspondences in p-adic families.

Abstract

We use results by Chenevier to interpolate the classical Jacquet-Langlands correspondence for Hilbert modular forms, which gives us an extension of Chenevier's results to totally real fields. From this we obtain an isomorphisms between eigenvarieties attached Hilbert modular forms and those attached to modular forms on a totally definite quaternion algebra over a totally real field of even degree.