Eisenstein series and the top degree cohomology of arithmetic subgroups of $S{L}_n/\mathbb{Q}$

Abstract

The cohomology $H^{\ast }(\Gamma ,E)$ of a torsion-free arithmetic subgroup $\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G}=S{L}_n$ may be interpreted in terms of the automorphic spectrum of $\Gamma$. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes $\{\mathsf{P}\}$ of associate proper parabolic $\mathbb{Q}$-subgroups of $\mathsf{G}$. Each summand $H_{\{\mathrm{P}\}}^{\ast }(\Gamma ,E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $\{\mathsf{P}\}$. The cohomology $H^{\ast }(\Gamma ,E)$ vanishes above the degree given by the cohomological dimension $\mathrm{cd}(\Gamma )=\frac{n(n-1)}{2}$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $\{\mathsf{P}\}$ for which the corresponding summand $H_{\{\mathsf{P}\}}^{\mathrm{cd}(\Gamma )}(\Gamma ,E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H_{\{\mathsf{Q}\}}^{\mathrm{cd}(\Gamma )}(\Gamma ,\mathbb{C})$. Finally, in the case of a principal congruence subgroup $\Gamma (q)$, $q=p^{\nu }>5$, $p\ge 3$ a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes $\{\mathsf{Q}\}$.

27 pages, no figures