{
  "abstract": "The key part of the current paper is the computation of boundary and Eisenstein cohomology of $GL_4({\\mathbb Z})$ with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of $SL_3({\\mathbb Z})$ and of $GL_3({\\mathbb Z})$ with coefficients in any highest weight representation. This is done in a simpler, faster and in a more structured way compared to \\cite{BHHM}. We state a duality for the boundary cohomology of $GL_m({\\mathbb Z})$ of the type of Serre's duality, where the dualizing sheaf is a power of the determinant representation. We refine this duality to a duality on the level of the spectral sequence for the boundary cohomology $E_\\infty^{p,q}$. We state it as a conjecture. However, all the computations, 30 different families of representations, satisfy this conjecture. We compute the Eisenstein cohomology of $GL_4({\\mathbb Z})$ with coefficients in the symmetric powers and their twist by the determinant representation. For several other representations, we compute the Eisenstein cohomology, based a few conjectures. Based on those conjectured, one can compute the Eisenstein cohomology in most of the cases. They will be included in the next version of the paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28869",
  "author_total": 1,
  "authors": [
    {
      "name": "Ivan Horozov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xb2v2m0cdq03.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28869",
  "fulltext_url": null,
  "id": "0xb2v2m0cdq03",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28869",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214394874",
  "pdf_url": "https://arxiv.org/pdf/2609.28869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xb2v2m0cdq03.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Computations of Cohomology of Arithmetic Groups, Part 1",
  "title_text": "Computations of Cohomology of Arithmetic Groups, Part 1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0xb2v2m0cdq03",
  "venue": "arXiv",
  "versions": []
}