The Jacquet Langlands correspondence via twisted descent

TL;DR

This paper extends the automorphic descent method to explicitly construct the Jacquet-Langlands correspondence for PGL(2), providing a new approach that complements classical trace formula and theta series methods.

Contribution

It introduces a novel extension of the automorphic descent method to explicitly realize the Jacquet-Langlands correspondence for PGL(2).

Findings

Recover classical Jacquet-Langlands correspondence via new explicit construction.

Extend automorphic descent method to a new setting.

Provide an alternative to trace formula and theta series approaches.

Abstract

The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.