{
  "abstract": "We combine the universal polynomial matrix identities of Larsen--Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of $\\text{SL}_2(\\mathcal{O}_{K,S})$ for every global function field $K$ and every nonempty finite set of places $S$. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over $\\mathbb{Z}$ that works simultaneously for all these rings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12705",
  "author_total": 1,
  "authors": [
    {
      "name": "Dong Quan Ngoc Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ergvn1x2drr7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12705",
  "fulltext_url": null,
  "id": "ergvn1x2drr7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12705",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7221576935",
  "pdf_url": "https://arxiv.org/pdf/2610.12705",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ergvn1x2drr7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12705",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12705"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Polynomial parametrization of $\\text{SL}_2$ over global function field rings",
  "title_text": "Polynomial parametrization of SL_2 over global function field rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/ergvn1x2drr7s",
  "venue": "arXiv",
  "versions": []
}