### Abstract

We extend the computations in [AGM11] to find the mod 2 homology in degree 1 of a congruence subgroup Γ of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is related to the cohomology of Γ 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 [AGM11]. Our method of computation was justified in [AGM12].

Original language | English (US) |
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Pages (from-to) | 4-22 |

Number of pages | 19 |

Journal | Journal of Number Theory |

Volume | 146 |

Issue number | C |

DOIs | |

State | Published - 2015 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Keywords

- Cohomology of arithmetic groups
- Galois representations
- Modular symbols
- Primary
- Secondary
- Steinberg module
- Voronoi complex

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## Cite this

*Journal of Number Theory*,

*146*(C), 4-22. https://doi.org/10.1016/j.jnt.2013.12.003