Cohomology of Congruence Subgroups of SL₄(Z). III

2009/03/18 by Avner Ash, Ash, Avner, Paul E. Gunnells +3 · 4 citations
Mathematics · #11F75 #65F05 #65F50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.0903.3201

openalex publication_date 2009/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In two previous papers [AGM1, AGM2] we computed cohomology groups H⁵(Γ₀ (N); \C) for a range of levels N, where Γ₀ (N) is the congruence subgroup of SL₄ (\Z) consisting of all matrices with bottom row congruent to (0,0,0,*) mod N. In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million non-zero entries. We also make two conjectures concerning the contributions to H⁵(Γ₀ (N); \C) for N prime coming from Eisenstein series and Siegel modular forms.

Cited by

• Mod 2 homology for GL(4) and Galois representations
• Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations
• On the cohomology of congruence subgroups of SL₄ (\Z)
• Resolutions of the Steinberg module for GL(n)

Related

• Cohomology of congruence subgroups of SL(4,Z)
• Cohomology of Congruence Subgroups of SL(4,\Z) II
• On the cohomology of congruence subgroups of SL₄ (\Z)
• Cohomology with Sym^g coefficients for congruence subgroups of SL₄(Z) and Galois representations
• Mod 2 homology for GL(4) and Galois representations