A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

Abstract

AbstractThe cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type {\mathrm{G}_{2}}, defined over a totally real number field. The construction of cuspidal automorphic representations of {\mathrm{GL}_{2}} with prescribed local and global properties is decisive in this context.