Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

2010/02/17 by Avner Ash, Ash, Avner, Paul E. Gunnells +4
Mathematics · #11F67 #11F75 #11F80 #11Y99 #20J06 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.GR #math.NT #msc:11F67 #msc:11F75 #msc:11F80 #msc:11Y99 #msc:20J06

paper · pdf · doi:10.48550/arxiv.1002.3385

arxiv created 2010/02/17 · openalex publication_date 2010/02/17 · arxiv updated 2010/02/26 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.

Citations

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

Related

• Torsions in Cohomology of SL₂(ℤ) and Congruence of Modular Forms
• Mod 2 homology for GL(4) and Galois representations
• Cohomology with Sym^g coefficients for congruence subgroups of SL₄(Z) and Galois representations
• Cohomology with twisted one-dimensional coefficients for congruence subgroups of SL(4,Z) and Galois representations
• Construction de représentations galoisiennes de torsion, d'après Peter Scholze