Vanishing of the cuspidal cohomology of congruence subgroups of \(\text{GL}_ n(\mathbb{Z})\)
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.
scientific article; zbMATH DE number 952630