{
  "abstract": "For an elliptic curve $E$ defined over a number field $K$ and $\\mathfrak{p}$ a prime of good reduction of $E/K$, we let $n_\\mathfrak{p}=\\#E(k_\\mathfrak{p})$ be the order of the group of points over the residue class field $k_\\mathfrak{p}$. Divisibility of $n_\\mathfrak{p}$ by a prime $\\ell$ depends on the splitting behavior of $\\mathfrak{p}$ in the $\\ell$-division field $K(E[\\ell](\\bar{K}))$ of $E$ and, generically, divisibility of $n_\\mathfrak{p}$ by different primes $\\ell_1$ and $\\ell_2$ are `independent events' on the set of primes $\\mathfrak{p}$ of good reduction. However, there do exist elliptic curves with `entangled division fields' for which infinitely many $n_\\mathfrak{p}$ are coprime to $\\ell_1$ and infinitely many $n_\\mathfrak{p}$ are coprime to $\\ell_2$, but no $n_\\mathfrak{p}$ is coprime to to $\\ell_1 \\ell_2$. For $\\ell_1=2$, we classify in group theoretical terms the entanglement giving rise to this behavior, and count up to isomorphism the number of `special $2\\ell$-division groups' exhibiting the phenomenon. A final section shows how such groups may be realized as the Galois group of $2\\ell$ division fields for elliptic curves $E/\\mathbf{Q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01204",
  "author_total": 4,
  "authors": [
    {
      "name": "Francesco Campagna"
    },
    {
      "name": "Nathan Jones"
    },
    {
      "name": "Francesco Pappalardi"
    },
    {
      "name": "Peter Stevenhagen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6x3x0d92k6f44.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01204",
  "fulltext_url": null,
  "id": "6x3x0d92k6f44",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01204",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6x3x0d92k6f44.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Entanglement for elliptic curve group orders",
  "title_text": "Entanglement for elliptic curve group orders",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/6x3x0d92k6f44",
  "venue": "arXiv",
  "versions": []
}