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Cohomology of congruence subgroups of \(\text{SL}(4,\mathbb Z)\). II. (English) Zbl 1213.11113

In Part I [J. Number Theory 94, No. 1, 181–212 (2002; Zbl 1006.11026)] the authors computed the cohomology groups \(H^5(\varGamma _0(N),\mathbb C)\), where \(\varGamma _{0}(N)\) is a certain congruence subgroup of \(\text{SL}(4,\mathbb Z)\), for a range of levels \(N\). In this note they update their earlier work by extending the range of levels (up to level 83) and describe cuspidal cohomology classes and additional boundary phenomena found since then. The cuspidal cohomology classes in this paper are the first cusp forms for \(\text{GL}(4)\) concretely constructed in terms of Betti cohomology.
In Part III [Math. Comput. 79, No. 271, 1811–1831 (2010; Zbl 1222.11069)] they update this earlier work by carrying it out for prime levels up to \(N=211\).

MSC:

11F75 Cohomology of arithmetic groups
11F80 Galois representations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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References:

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