Global Jacquet-Langlands correspondence for division algebras in characteristic p

TL;DR

This paper establishes a comprehensive global Jacquet-Langlands correspondence between automorphic representations of GL(n) and division algebras over global fields of positive characteristic, extending local results to a global setting.

Contribution

It proves the existence of an injective map linking automorphic square integrable representations of division algebra groups to those of GL(n), and characterizes its image in positive characteristic.

Findings

Existence of an injective map between automorphic representations of division algebras and GL(n).

Characterization of the image of this map.

Results imply multiplicity one and strong multiplicity one theorems.

Abstract

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. As a consequence we get multiplicity one and strong multiplicity one theorems for the multiplicative group of D.