{
  "abstract": "Given a maximal order $\\mathfrak{O}$ in a rational definite quaternion algebra, and a prime $\\ell$ coprime to the discriminant of $\\mathfrak{O}$, this paper considers subsets of $\\mathfrak{O}$ consisting of elements with $\\ell$-power norms that together generate $\\mathfrak{O}$ as a $\\mathbb{Z}$-algebra. We prove two theorems about the existence of such sets: the first states that $\\mathfrak{O}$ is generated by elements of norm $\\ell^k$ for any $k$ larger than an explicit bound, and the second states that there is a generating set for $\\mathfrak{O}$ consisting of at most three elements, each with norm a power of $\\ell$. We discuss implications for the study of supersingular isogeny graphs. As steps towards these theorems, for quaternion orders $\\mathcal{O}$ that are not necessarily maximal, we also prove structural results about the algebra of Brandt matrices for $\\mathcal{O}$ and explicit bounds on the coefficients of the theta function of $\\mathcal{O}$. Computational experiments are also discussed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10139",
  "author_total": 9,
  "authors": [
    {
      "name": "Kirsten Eisenträger"
    },
    {
      "name": "Eyal Z. Goren"
    },
    {
      "name": "Annamaria Iezzi"
    },
    {
      "name": "Harun Kir"
    },
    {
      "name": "Eda Kırımlı"
    },
    {
      "name": "Jonathan R. Love"
    },
    {
      "name": "William E. Mahaney"
    },
    {
      "name": "Jennifer Park"
    },
    {
      "name": "Maria Sabitova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6yynv9t0469c9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10139",
  "fulltext_url": null,
  "id": "6yynv9t0469c9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10139",
  "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": "W7220976970",
  "pdf_url": "https://arxiv.org/pdf/2610.10139",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6yynv9t0469c9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10139",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10139"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generating sets for maximal orders in rational quaternion algebras",
  "title_text": "Generating sets for maximal orders in rational quaternion algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/6yynv9t0469c9",
  "venue": "arXiv",
  "versions": []
}