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

Joachim Schwermer

Vol. 306 (2020), No. 1, 321–355

DOI: 10.2140/pjm.2020.306.321

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 $GL\left(2,D\right)∕k$ has $k$-rank one; it is an inner form of the split $k$-group $GL\left(2d\right)∕k$. We construct automorphic representations $\pi$ of $GL\left(2d\right)∕k$ which occur nontrivially in the discrete spectrum of $GL\left(2d,k\right)$ and which have specific local components at archimedean as well as nonarchimedean places of $k$ so that there exist automorphic representations ${\pi }^{\prime }$ of $GL\left(2,D\right)\left({\mathbb{𝔸}}_k\right)$ with $\Xi \left({\pi }^{\prime }\right)=\pi$ 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 ${\pi }^{\prime }$ of $GL\left(2,D\right)\left({\mathbb{𝔸}}_k\right)$ 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