The residual Eisenstein cohomology of \(\mathrm{Sp}_4\) over a totally real number field (Q2847192)

The introductory section of the paper under review does an excellent job of presenting its subject (both from a mathematical and a historical perspective) and the reviewer does not feel he could add anything to it. Let us just mention for quick reference that the present paper is a generalisation of the investigations of \textit{J. Schwermer} about the Eisenstein cohomology of the symplectic group of rank 2 over \(\mathbb{Q}\) [Compos. Math. 58, 233--258 (1986; Zbl 0596.10029)] to the case where \(\mathbb{Q}\) is replaced by a general number field.NEWLINENEWLINEFor more details on the contents, let us quote the authors' own summary of their work:NEWLINENEWLINE``Let \(G= \mathrm{Sp}_4/k\) be the \(k\)-split symplectic group of \(k\)-rank 2, where \(k\) is a totally real number field. In this paper we compute the Eisenstein cohomology of \(G\) with respect to any finite-dimensional, irreducible, \(k\)-rational representation \(E\) of \(G_\infty = R_{k/\mathbb{Q}}G(\mathbb{R})\), where \(R_{k/\mathbb{Q}}G\) denotes the restriction of scalars from \(k\) to \(\mathbb{Q}\). This approach is based on the work of Schwermer regarding the Eisenstein cohomology for \(\mathrm{Sp}_4/\mathbb{Q}\), Kim's description of the residual spectrum of \(\mathrm{Sp}_4\), and the Franke filtration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group \(G\), the Franke filtration with respect to the cuspidal support, and give a precise description of the filtration 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-integrable, that is, represented by a non-square-integrable residue of an Eisenstein series.''