Eisenstein series and the top degree cohomology of arithmetic subgroups of SL n /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 = SL n {\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 ) = 1 2 ⁢ 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 = p ν > 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.