{
  "abstract": "In this paper, we obtain an exact sequence connecting $H_2(\\rm{SL}_2(\\mathcal{O}_{K,S}), \\mathbb{Z})$ to $H_2(\\rm{SL}_2(\\mathcal{O}_{K,T}), \\mathbb{Z})$, where $\\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\\rm{SL}_2(\\mathcal{O}_{K,S}), \\mathbb{Z})$ with the second $K$-group $K_2(\\mathcal{O}_{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\\rm{SL}_2(\\mathcal{O}_{K,S}), \\mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28677",
  "author_total": 4,
  "authors": [
    {
      "name": "P. H. Amorim"
    },
    {
      "name": "I. V. Picinini"
    },
    {
      "name": "B. R. Ramos"
    },
    {
      "name": "T. Verissimo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f2vd7wsxrn2dy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28677",
  "fulltext_url": null,
  "id": "f2vd7wsxrn2dy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28677",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.KT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214284430",
  "pdf_url": "https://arxiv.org/pdf/2609.28677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f2vd7wsxrn2dy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.28677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Schur multiplier of $\\rm{SL}_2$ and Dedekind zeta-functions over $S$-integers",
  "title_text": "The Schur multiplier of SL_2 and Dedekind zeta-functions over S-integers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/f2vd7wsxrn2dy",
  "venue": "arXiv",
  "versions": []
}