The Eisenstein motive for the cohomology of GSp₂(ℤ)

Abstract

In his paper [4], Gerard van der Geer discusses the Eisenstein cohomology with coefficients in a sheaf $\tilde{M}$, which is obtained from a representation for the group \( \tilde\Gamma=\GSp_g(\mathbb Z) \). Since we have an arithmetic interpretation of this sheaf, we can endow these cohomology groups with the structure of a mixed motive. A certain part of this cohomology is the compactly supported Eisenstein cohomology and van der Geer determines the structure of this compactly supported Eisenstein motive in the case $g=2$ and a regular coefficient system [4], Cor.~10.2). At the end of this note we compute this part of the cohomology for an arbitrary coefficient system, again in the case $g=2$.