{
  "abstract": "We investigate how Heegner points detect the finiteness of the $p$-primary part of Tate-Shafarevich groups of semi-stable elliptic curves over the rationals of \\emph{arbitrary} rank with any good reduction prime $p \\geq 5$ with irreducible mod $p$ representation and partially recover Çiperiani--Wiles' theorem on solvable points on genus one curves. As a key ingredient of the proof of these results, we also give a concise proof of Kolyvagin's conjecture for semi-stable elliptic curves with supersingular reduction with a relevant choice of an imaginary quadratic field. Our approach is independent of the cyclotomic Iwasawa main conjecture for elliptic curves with supersingular reduction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15328",
  "author_total": 1,
  "authors": [
    {
      "name": "Chan-Ho Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2hbk9qbqgc3d7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15328",
  "fulltext_url": null,
  "id": "2hbk9qbqgc3d7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15328",
  "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": "W7213268875",
  "pdf_url": "https://arxiv.org/pdf/2609.15328",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2hbk9qbqgc3d7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.15328",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15328"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture",
  "title_text": "On the arithmetic of Tate-Shafarevich groups via Kolyvagin's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2hbk9qbqgc3d7",
  "venue": "arXiv",
  "versions": []
}