{
  "abstract": "Let $E$ be an elliptic curve over $\\mathbb{Q}_p$. We study the field $\\mathbb{Q}_p(E[p])$ generated by the $p$-torsion points of $E$. When $E$ has good reduction, we determine not only $\\mathbb{Q}_p(E[p])$ but also the field $\\mathbb{Q}_p(P)$ for every point $P\\in E[p]$. This classification yields criteria for the existence of $p$-torsion over unramified and Lubin--Tate extensions of $\\mathbb{Q}_p$. As a global application, let $E/\\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction. Define $f_p(E)$ to be the multiplicative order of $a_p(E)$ modulo $p$ in the ordinary case, and set $f_p(E)=p^2-1$ in the supersingular case. We prove that $E(K)[p]=0$ for every number field $K/\\mathbb{Q}$ with $[K:\\mathbb{Q}]<f_p(E)$. When $E$ has bad reduction, we describe the ramified part of $\\mathbb{Q}_p(E[p])/\\mathbb{Q}_p$. For every reduction type, we determine the maximal upper ramification break of $\\mathbb{Q}_p(E[p])/\\mathbb{Q}_p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06554",
  "author_total": 2,
  "authors": [
    {
      "name": "Yoshiyasu Ozeki"
    },
    {
      "name": "Manabu Yoshida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmxe958fzq1rf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06554",
  "fulltext_url": null,
  "id": "dmxe958fzq1rf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06554",
  "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": "W7212004931",
  "pdf_url": "https://arxiv.org/pdf/2609.06554",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmxe958fzq1rf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06554",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06554"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Fields of definition of $p$-torsion points of elliptic curves and their ramification",
  "title_text": "Fields of definition of p-torsion points of elliptic curves and their ramification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/dmxe958fzq1rf",
  "venue": "arXiv",
  "versions": []
}