{
  "abstract": "In this paper, we show that $\\operatorname{SL}_2(\\mathbb{F}_{13})$ and $\\operatorname{SL}_2(\\mathbb{F}_{19})$ are Galois groups of totally real extensions of $\\mathbb{Q}$. For each of these primes $\\ell$, we find a polynomial of degree $\\ell+1$ whose splitting field has Galois group $\\operatorname{PSL}_2(\\mathbb{F}_\\ell)$. These fields satisfy Böge's criterion and the associated central embedding problem has a proper solution with Galois group $\\operatorname{SL}_2(\\mathbb{F}_\\ell)$. One can choose the resulting fields totally real via a quadratic twist. The degree $14$ polynomial is found through a genus one Hurwitz family of degree 14 covers. The degree 20 polynomial is found using Yang's modular equations for the Shimura curve $X_6^*(1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03946",
  "author_total": 1,
  "authors": [
    {
      "name": "Emir Eray Karabiyik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/60323y93deta8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03946",
  "fulltext_url": null,
  "id": "60323y93deta8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03946",
  "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.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7208734067",
  "pdf_url": "https://arxiv.org/pdf/2609.03946",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/60323y93deta8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03946",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03946"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The groups $\\operatorname{SL}_2(\\mathbb{F}_{13})$ and $\\operatorname{SL}_2(\\mathbb{F}_{19})$ are Galois over $\\mathbb{Q}$",
  "title_text": "The groups SL_2(𝔽_13) and SL_2(𝔽_19) are Galois over ℚ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/60323y93deta8",
  "venue": "arXiv",
  "versions": []
}