Eisenstein series, cohomology of arithmetic groups, and automorphic L-functions at half integral arguments

Abstract

Abstract The automorphic cohomology of a reductive ℚ-group G, defined in terms of the automorphic spectrum of G, captures essential analytic aspects of the arithmetic subgroups of G and their cohomology. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic ℚ-subgroups of G, forms the Eisenstein cohomology which is a natural complement to the cuspidal cohomology. We show that non-trivial Eisenstein cohomology classes can only arise if the point of evaluation features a `half-integral' property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.