### Abstract

In two previous papers we computed cohomology groups H^{5}(Γ_{0}(N); ℂ) for a range of levels N, where Γ_{0}(N) is the congruence subgroup of SL_{4}(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 nonzero entries. We also make two conjectures concerning the contributions to H^{5}(Γ_{0}(N); [ℂ]) for N prime coming from Eisenstein series and Siegel modular forms.

Original language | English (US) |
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Pages (from-to) | 1811-1831 |

Number of pages | 21 |

Journal | Mathematics of Computation |

Volume | 79 |

Issue number | 271 |

DOIs | |

State | Published - Jul 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Keywords

- Automorphic forms
- Cohomology of arithmetic groups
- Eisenstein cohomology
- Hecke operators
- Paramodular group
- Siegel modular forms
- Smith normal form
- Sparse matrices

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## Cite this

Ash, A., Gunnells, P. E., & McConnell, M. (2010). Cohomology of congruence subgroups of SL

_{4}(ℤ). III.*Mathematics of Computation*,*79*(271), 1811-1831. https://doi.org/10.1090/S0025-5718-10-02331-8