The residual Eisenstein cohomology of $Sp_{4}$ over a totally real number field

Abstract

Let G = Sp4=k be the k-split symplectic group of k-rank 2, where k is a totally real numbereld. In this paper we compute the Eisenstein cohomology of G with respect to anynite{ dimensional, irreducible, k-rational representation E of G1 = Rk=QG( R), where Rk=Q denotes the restriction of scalars from k to Q. The approach is based on the work of Schwermer regarding the Eisenstein cohomology for Sp4=Q, Kim's description of the residual spectrum of Sp4, and the Frankeltration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group G, the Frankeltration with respect to the cuspidal support, and give a precise description of theltration quotients in terms of induced representations. This is then used as a prerequisite for the explicit computation of the Eisenstein cohomology. The special focus is on the residual Eisenstein cohomology. Under a certain compatibility condition for the coefficient system E and the cuspidal support, we prove the existence of non{trivial residual Eisenstein cohomology classes, which are not square{integrabl e, that is, represented by a non{ square{integrabl residue of an Eisenstein series.