Global Jacquet-Langlands Correspondence for Division Algebras in Characteristic p

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 representations of D-x 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 D-x.