Vanishing of the cuspidal cohomology of congruence subgroups of \(\text{GL}_ n(\mathbb{Z})\)
type: Item
instance of: scholarly article
title: Vanishing of the cuspidal cohomology of congruence subgroups of \(\text{GL}_ n(\mathbb{Z})\)
zbMATH Open document ID: 0866.11038
DOI: 10.1007/BF01445250
author: Stéfane Fermigier
published in: Mathematische Annalen
publication date: 1996-12-19T00:00:00Z
full work available at URL: https://eudml.org/doc/165453
review text: A central problem in the theory of automorphic forms is the determination of the cohomology groups \(H^\bullet(\Gamma,\mathbb{C})\) for an arithmetic group \(\Gamma\). By general principles this cohomology can be computed by differentiable cohomology of the ambient algebraic group. This gives rise to a subgroup \(H^\bullet_{\text{cusp}} (\Gamma,\mathbb{C})\), the cuspidal cohomology, stemming from the contribution of cups forms. The cuspidal cohomology is a direct summand [\textit{A. Borel}, \textit{J.-P. Labesse}, and \textit{J. Schwermer}, Compos. Math. 102, 1-40 (1996; Zbl 0853.11044)]. The other summands come from Levi-components of lower dimension, therefore the cuspidal cohomology is the most interesting part of the cohomology of an arithmetic group. In the literature one mainly finds nonvanishing results giving conditions on an arithmetic group \(\Gamma\) which ensure the existence of a subgroup \(\Gamma'\) of finite index with \(H^\bullet_{\text{cusp}} (\Gamma',\mathbb{C})\neq 0\). For a nice survey see \textit{J. Schwermer} [Cohomology of arithmetic groups and automorphic forms, Proc. Conf., Luminy/Fr. 1989, Lect. Notes Math. 1447, 1-29 (1990; Zbl 0715.11028)]. The present paper is concerned with the inverse problem. For the arithmetic groups \(\Gamma_1(N)\) explicit bounds are given for the conductor \(N\) below which there is no cuspidal cohomology. One for example gets that for \(n\leq 23\): \[ H^\bullet_{\text{cusp}} (GL_n(\mathbb{Z}),\mathbb{C})=0. \] The proof is based on ``explicit formulas'' à la Weil for automorphic \(L\)-functions giving bounds for the conductor.
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 952630
zbMATH Keywords: cohomology groups
zbMATH Keywords: cuspidal cohomology
zbMATH Keywords: arithmetic groups
zbMATH Keywords: automorphic \(L\)-functions
reviewed by: Anton Deitmar
MaRDI profile type: Publication
cites work: Computations of cuspidal cohomology of congruence subgroups of SL(3,\({\mathbb{Z}})\)
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cites work: Unitary representations of Gl(n,R) with non-trivial (g,K)-cohomology
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description: scientific article; zbMATH DE number 952630