Ash, Avner; Doud, Darrin; Pollack, David Galois representations with conjectural connections to arithmetic cohomology. (English) Zbl 1023.11025 Duke Math. J. 112, No. 3, 521-579 (2002). In [Duke Math. J. 54, 179-230 (1987; Zbl 0641.10026)], J.-P. Serre published a conjecture relating continuous \(2\)-dimensional absolutely irreducible Galois representations (in characteristic \(p\)) to the reductions modulo \(p\) of modular forms. In a joint paper [Duke Math. J. 105, 1-24 (2000; Zbl 1015.11018)], the first author and W. Sinnott stated another conjecture that relates niveau \(1\) Galois representations of arbitrary dimension \(n\) and cohomology groups of congruence subgroups of \(\text{GL}(n,\mathbb Z)\). In the case \(n = 2\), this conjecture of Ash and Sinnott is closely related to Serre’s conjecture. An expanded version of the Ash-Sinnott conjecture is given in the present paper that includes Galois representations of higher niveau. Computational evidence for the last conjecture in the case \(n = 3\) is presented (in the form of Galois representations for which the coefficients of the characteristic polynomials of the images of the Frobenius match the Hecke eigenvalues for the corresponding cohomology of congruence subgroups). Using the results of A. Ash and Pham Huu Tiep [J. Algebra 222, 376-399 (1999; Zbl 0949.11032)], the authors also show that the conjecture holds for the symmetric squares of certain \(2\)-dimensional Galois representations. Some computational tests for the conjecture in the case \(n = 4\) have recently been done in A. Ash, P. E. Gunnells and M. McConnell [J. Number Theory 94, 181-212 (2002; Zbl 1006.11026)]. Reviewer: Pham Huu Tiep (Gainesville) Cited in 10 ReviewsCited in 34 Documents MSC: 11F75 Cohomology of arithmetic groups 11F80 Galois representations Keywords:cohomology of arithmetic groups; Galois representations Citations:Zbl 0641.10026; Zbl 1015.11018; Zbl 0949.11032; Zbl 1006.11026 Software:PARI/GP × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] G. Allison, A. Ash, and E. Conrad, Galois representations, Hecke operators, and the mod-\(p\) cohomology of \({\GL}(3,\mathbb{Z})\) with twisted coefficients, Experiment. Math. 7 (1998), 361–390. · Zbl 0923.11083 · doi:10.1080/10586458.1998.10504381 [2] A. Ash, Galois representations attached to mod \(p\) cohomology of {\(\GL(n,\mathbb{Z})\) , Duke Math. J. 65 (1992), 235–255.} · Zbl 0774.11024 · doi:10.1215/S0012-7094-92-06510-0 [3] A. Ash and M. McConnell, Experimental indications of three-dimensional Galois representations from the cohomology of {\({\mathrm SL}(3,\mathbb{Z})\) , Experiment. 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