Vol. 306, No. 1, 2020

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Nonvanishing square-integrable automorphic cohomology classes: the case GL(2) over a central division algebra

Joachim Schwermer

Vol. 306 (2020), No. 1, 321–355
Abstract

Let k be a totally real algebraic number field, and let D be a central division algebra of degree d over k. The connected reductive algebraic k-group GL(2,D)k has k-rank one; it is an inner form of the split k-group GL(2d)k. We construct automorphic representations π of GL(2d)k which occur nontrivially in the discrete spectrum of GL(2d,k) and which have specific local components at archimedean as well as nonarchimedean places of k so that there exist automorphic representations π of GL(2,D)(𝔸k) with Ξ(π) = π under the Jacquet–Langlands correspondence. These requirements depend on the finite set V D of places of k at which D does not split, and on the quest to construct representations π of GL(2,D)(𝔸k) which either represent cuspidal cohomology classes or give rise to square-integrable classes which are not cuspidal, that is, are eventually represented by a residue of an Eisenstein series. The demand for cohomological relevance gives strong constraints at the archimedean components.

Keywords
cohomology of arithmetic groups, square-integrable automorphic representations, general linear group over a central division algebra
Mathematical Subject Classification 2010
Primary: 11F70, 11F75
Secondary: 22E40, 22E55
Milestones
Received: 19 October 2018
Accepted: 30 December 2019
Published: 14 June 2020
Authors
Joachim Schwermer
Faculty of Mathematics
University of Vienna
Vienna
Austria
Max Planck Institute for Mathematics
Bonn
Germany