# The Trace Formula and the Proof of the Global Jacquet-Langlands Correspondence

## TL;DR

This paper surveys the proof of the global Jacquet-Langlands correspondence between GL_n and its inner forms, emphasizing the trace formula approach and providing detailed background and context.

## Contribution

It offers a comprehensive overview of the proof of the global Jacquet-Langlands correspondence in characteristic zero, based on lecture notes and detailed explanations.

## Key findings

- Clarifies the role of the trace formula in the proof
- Details the objects and results involved in the correspondence
- Provides a pedagogical introduction to the proof process

## Abstract

This is a survey of the proof of the global Jacquet-Langlands correspondence between GL_n and general inner forms, in characteristic zero. The proof is given after a long introduction recalling in some detail the objects and results involved. The paper is based on the notes of the lectures I gave at the doctoral school in CIRM 2016 (Jean-Morlet Chair) organized by Dipendra Prasad, Volker Heiermann and Fiona Murnaghan. It appeared in 2018 in the volume Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms (LNM 2221).