Labesse, J.-P.; Schwermer, J. On liftings and cusp cohomology of arithmetic groups. (English) Zbl 0581.10013 Invent. Math. 83, 383-401 (1986). This paper confronts the difficult problem of determining the cuspidal part of the cohomology of an arithmetic group. The object here is to show that for certain algebraic number fields E with ring of integers \({\mathcal O}_ E\) there exist congruence subgroups of \(SL_ 2({\mathcal O}_ E)\) and \(SL_ 3({\mathcal O}_ E)\) with non-vanishing cuspidal cohomology. The condition on E is arithmetic and quite weak. The construction used here is based on Langlands functorality and employs both the Saito-Shintani- Langlands theory of base change for \(GL_ 2\) and the Gelbart-Jacquet lift from \(GL_ 2\) to \(GL_ 3\). The proof also uses a version of the Selberg trace formula and several other related interesting techniques. Reviewer: S.J.Patterson Cited in 28 Documents MSC: 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory Keywords:arithmetic group; congruence subgroups; cuspidal cohomology; Langlands functorality; Saito-Shintani-Langlands theory; base change for \(GL_ 2\); Gelbart-Jacquet lift; Selberg trace formula PDFBibTeX XMLCite \textit{J. P. Labesse} and \textit{J. Schwermer}, Invent. Math. 83, 383--401 (1986; Zbl 0581.10013) Full Text: DOI EuDML References: [1] Borel, A.: Stable real cohomology of arithmetic groups II. In: Hano, J., et al. (eds.) Manifolds and Lie groups. Progress in Maths., vol. 14, pp. 21-55. Boston-Basel-Stuttgart: Birkhäuser 1981 · Zbl 0483.57026 [2] Borel, A.:Automorphic L-functions. In: Proc. Symp. Pure Maths., vol. 33, part 2, pp. 27-62. 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