Automorphic Forms, Cohomology and CAP Representations. The Case $GL_2$ over a definite quaternion algebra

TL;DR

This paper characterizes the cohomology of $GL_2$ over a definite quaternion algebra, revealing the existence of special automorphic forms including CAP-representations, and explores implications for $L$-value rationality.

Contribution

It provides a complete description of cuspidal and Eisenstein cohomology for $GL_2$ over a quaternion algebra, highlighting the existence of CAP-representations with cohomology in degree 1.

Findings

Identification of residual and cuspidal automorphic forms with cohomology in degree 1.

Existence of CAP-representations satisfying Strong Multiplicity One.

Non-vanishing of intertwining operators as a basis for future $L$-value studies.

Abstract

In this paper we fully describe the cuspidal and the Eisenstein cohomology of the group $G=GL_2$ over a definite quaternion algebra $D/\Q$. Functoriality is used to show the existence of residual and cuspidal automorphic forms, having cohomology in degree 1. The latter ones turn out to be CAP-representations, though $G$ satisfies Strong Multiplicity One. A non-vanishing result on intertwining operators of induced representations will serve as a starting point for further investigations concerning rationality of critical $L$-values.