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

DOI10.1515/CRELLE-2021-0022arXiv2003.04611MaRDI QIDQ6336448

Joachim Schwermer

Publication date: 10 March 2020

Abstract: The cohomology $H^{*}(Gamma,E)$ of a torsion-free arithmetic subgroup $Gamma$ of the special linear $mathbbQ$-group $mathsfG=S{L}_n$ 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 $mathsfP$ of associate proper parabolic $mathbbQ$-subgroups of $mathsfG$. Each summand $H_{mathrmP}^{*}(Gamma,E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $mathsfP$. The cohomology $H^{*}(Gamma,E)$ vanishes above the degree given by the cohomological dimension $mathrmcd(Gamma)=fracn(n-1)2$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $mathsfP$ for which the corresponding summand $H_{mathrmmathsfP}^{mathrmcd(Gamma)}(Gamma,E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H_{mathrmmathsfQ}^{mathrmcd(Gamma)}(Gamma,mathbbC)$. Finally, in the case of a principal congruence subgroup $Gamma(q)$, $q=p^u>5$, $pgeq3$ 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 $mathsfQ$.

Mathematics Subject Classification ID

Discrete subgroups of Lie groups (22E40) Special values of automorphic (L)-series, periods of automorphic forms, cohomology, modular symbols (11F67) Cohomology of arithmetic groups (11F75)

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