{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_NUMBER_THEORY-0109",
  "title": "A Positive-Rank Elliptic Curve with No Dense Prime",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An AIM problem asks whether every positive-rank elliptic curve over ℚ has a prime p for which its rational points are dense in its p-adic points. We give a negative answer. For E: y² = x³ − 1516563 and P = (6403/9, 511280/27), an unconditional full 2-descent and saturation certify E(ℚ) = ℤP. A rational 3-isogeny places the reduction of every rational point in a subgroup of index 3 at every good prime, while separate exact arguments exclude density at the three bad primes 2, 3, and 79. Thus the dense-prime set is empty. We also record an exact finite local criterion: density is equivalent to surjectivity on a fixed finite quotient of the Néron filtration, and, at a good odd prime, on 𝓔(ℤ/p²ℤ).",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.NT"
  ],
  "keywords": [
    "elliptic curves",
    "p-adic density",
    "Mordell–Weil groups",
    "isogenies",
    "Néron models",
    "elliptic curve",
    "Mordell–Weil group",
    "isogeny",
    "formal group",
    "AIM-ALGEBRAIC_NUMBER_THEORY-0109",
    "open mathematics",
    "mathematical proof",
    "math.NT"
  ],
  "manuscript_version_date": "2026-09-01",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0109/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-number-theory-0109/paper.pdf?v=f0b485c06a5e",
  "doi": "10.5281/zenodo.22245533",
  "zenodo_record_url": "https://zenodo.org/records/22245533",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
    "paper.pdf": {
      "sha256": "f0b485c06a5e441127cafd3c1e23deac1b26076f4c1fc6be5627735243064390"
    },
    "source.zip": {
      "sha256": "e43e05a385c85b41dbb6e4ba70467edcd4ab67d2dd4d903e05988b2a8624ee66"
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    "verification_report.md": {
      "sha256": "5e2128b212a164c70b63517f8c8a33a9fba2879d6ba6f44c010d2d9e53de049b"
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  },
  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}