{
  "abstract": "Let $E/K$ be an elliptic curve over a number field $K$, and let $[C]\\in H^1(K,E(\\overline{K}))$ be a homogeneous space under $E$. Suppose that $C$ has period $n$ and index $d$. O'Neil asked whether one can always choose a lift of $[C]$ to $H^1(K,E[n])$ whose period-index obstruction has order exactly $d/n$. We give a negative answer to this question by constructing an explicit family of examples.Taking $K=\\mathbb{Q}(ζ_8)$, we prove that there exist infinitely many pairwise non-isomorphic elliptic curves $E/K$ such that each $E$ admits infinitely many homogeneous spaces $[C]\\in H^1(K,E(\\overline{K}))$ with period $8$ and index $16$, but the period-index obstruction of every lift of $[C]$ to $H^1(K,E[8])$ has order exactly $8$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29288",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoguang Shang"
    },
    {
      "name": "Cheng Niu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rcvhd1m4a26y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29288",
  "fulltext_url": null,
  "id": "0rcvhd1m4a26y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29288",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204882094",
  "pdf_url": "https://arxiv.org/pdf/2608.29288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rcvhd1m4a26y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves",
  "title_text": "Counterexamples to O'Neil's Period-Index Problem on Elliptic Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/0rcvhd1m4a26y",
  "venue": "arXiv",
  "versions": []
}