{
  "abstract": "Let $F$ be a totally real Galois number field of degree $g\\leq5$ and let $\\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\\ell$, we prove that there are $\\gg_{F,E,\\ell}\\frac{X^{1/(2g)}}{\\log X}$ totally negative square classes $d\\in F^\\times/(F^\\times)^2$ with $\\left|\\mathrm{N}_{F/\\mathbb{Q}}\\bigl(D(F(\\sqrt{d})/F)\\bigr)\\right|<X $ for which $\\operatorname{Sel}_\\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\\ell$-part of the Shafarevich--Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23847",
  "author_total": 2,
  "authors": [
    {
      "name": "Olivia Beckwith"
    },
    {
      "name": "Tùng Hoàng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4cjj54hzyrjtj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23847",
  "fulltext_url": null,
  "id": "4cjj54hzyrjtj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23847",
  "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.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214042675",
  "pdf_url": "https://arxiv.org/pdf/2609.23847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4cjj54hzyrjtj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.23847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Selmer groups of CM-twists of elliptic curves",
  "title_text": "Selmer groups of CM-twists of elliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/4cjj54hzyrjtj",
  "venue": "arXiv",
  "versions": []
}