{
  "abstract": "Let $E$ be a rational elliptic curve. We generalize a theorem due to Avcı\\cite{AVCI2026153}, which asserts that for any quadratic field $K$ and prime $p>5$, the equality $E(K)_{\\mathrm{tors}} = E(L)_{\\mathrm{tors}}$ holds for every $\\mathbb{Z}_p$-extension $L/K$. In this paper, we consider the setting where the $\\mathbb{Z}_p$-extension $L$ is replaced by the compositum $K_{\\infty}$ of all $\\mathbb{Z}_p$-extensions of $K$. Under this new setting, we prove the analogous statement for $p=5$, and further provide some results for the remaining primes $p=3$ and $p=2$, where we also try to study the growth pattern of torsion groups. Moreover, we completely classify all triples $(E,K,p)$ for which $E(K_\\infty)_{\\mathrm{tors}}$ is infinite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13514",
  "author_total": 1,
  "authors": [
    {
      "name": "Haidong Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6nr5mcmjmmsdx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13514",
  "fulltext_url": null,
  "id": "6nr5mcmjmmsdx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13514",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7168921625",
  "pdf_url": "https://arxiv.org/pdf/2607.13514",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6nr5mcmjmmsdx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.13514",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13514"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Torsion groups of rational elliptic curves over $\\mathbb{Z}_p$-extensions of quadratic fields: the $p\\le 5$ case",
  "title_text": "Torsion groups of rational elliptic curves over ℤ_p-extensions of quadratic fields: the p≤ 5 case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6nr5mcmjmmsdx",
  "venue": "arXiv",
  "versions": []
}