On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1

TL;DR

This paper develops a $p$-adic extension of the Jacquet-Langlands correspondence, providing explicit examples of overconvergent automorphic forms of weight 1 linked to classical modular forms, combining experimental and theoretical approaches.

Contribution

It introduces a novel $p$-adic extension of the Jacquet-Langlands correspondence and constructs explicit examples of overconvergent forms of weight 1.

Findings

Explicit example of an overconvergent automorphic form of weight 1

Correspondence established between overconvergent and classical forms

Combination of experimental and theoretical methods used

Abstract

We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which corresponds to a classical modular form of weight~1, using both experimental and theoretical methods.