Schwermer, Joachim (2020) Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$. MPIM Preprint Series 2020 (46).
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Abstract
The cohomology $H^*(\Gamma, E) $ of a torsion-free arithmetic subgroup $\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ 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 $H^*_{\mathrm{\{P\}}}(\Gamma, E)$ 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^*(\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 $H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{P}\}}}(\Gamma, E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{Q}\}}}(\Gamma, \mathbb{C})$.
Finally, in the case of a principal congruence subgroup $\Gamma(q)$, $q = p^{\nu} > 5$, $p\geq 3$ 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 $\{\mathsf{Q}\}$.
| Item Type: | MPIM Preprint |
|---|---|
| Subjects: | 1 Discrete mathematics / algebra > 11-XX Number theory 1 Discrete mathematics / algebra > 22-XX Topological groups, Lie groups |
| Divisions: | Research > Preprints |
| Depositing User: | This Admin |
| Date Deposited: | 17 Nov 2020 13:28 |
| Last Modified: | 17 Nov 2020 13:29 |
| URI: | https://archive.mpim-bonn.mpg.de/id/eprint/243 |
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