Eisenstein series and the top degree cohomology of arithmetic subgroups of SLn/ℚ

Abstract

Abstract The cohomology H*⁢(Γ,E){H^{*}(\Gamma,E)} of a torsion-free arithmetic subgroup Γ of the special linear ℚ{\mathbb{Q}}-group 𝖦=SLn{\mathsf{G}={\mathrm{SL}}_{n}} may be interpreted in terms of the automorphic spectrum of Γ. 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 {𝖯}{\{\mathsf{P}\}} of associate proper parabolic ℚ{\mathbb{Q}}-subgroups of 𝖦{\mathsf{G}}. Each summand H{P}*⁢(Γ,E){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 {𝖯}{\{\mathsf{P}\}}. The cohomology H*⁢(Γ,E){H^{*}(\Gamma,E)} vanishes above the degree given by the cohomological dimension cd⁢(Γ)=12⁢n⁢(n-1){\mathrm{cd}(\Gamma)=\frac{1}{2}n(n-1)}. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes {𝖯}{\{\mathsf{P}\}} for which the corresponding summand H{𝖯}cd⁢(Γ)⁢(Γ,E){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{𝖰}cd⁢(Γ)⁢(Γ,ℂ){H^{\mathrm{cd}(\Gamma)}_{\mathrm{\{\mathsf{Q}\}}}(\Gamma,\mathbb{C})}. Finally, in the case of a principal congruence subgroup Γ⁢(q){\Gamma(q)}, q=pν>5{q=p^{\nu}>5}, p≥3{p\geq 3} a prime, we give lower bounds for the size of these spaces. In addition, for certain associate classes {𝖰}{\{\mathsf{Q}\}}, there is a precise formula for their dimension.