On some arithmetic properties of automorphic forms of GL(m) over a division algebra

In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e., D=F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke-Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G' to statements on automorphic forms of its split form using the global Jacquet-Langlands correspondence developed by Badulescu and Badulescu-Renard. Beside that we prove that the local version of the Jacquet-Langlands transfer at an archimedean place preserves the property of being cohomological.

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