Mod 2 homology for GL(4) and Galois representations

2013/04/21 by Avner Ash, Ash, Avner, Paul E. Gunnells +3
Mathematics · #11F67 #11F75 #20E42 #20J06 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.NT #msc:11F67 #msc:11F75 #msc:20E42 #msc:20J06

paper · pdf · doi:10.48550/arxiv.1304.5684

openalex publication_date 2013/04/21 · arxiv created 2013/06/13 · arxiv updated 2013/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We extend the computations in [AGM4] to find the mod 2 homology in degree 1 of a congruence subgroup Gamma of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is closely related to the cohomology of Gamma with F₂ coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in [AGM4]. Our method of computation was justified in [AGM5].

Citations

• Cohomology of Congruence Subgroups of SL₄(Z). III

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