{
  "schema_version": 1,
  "problem_number": "OWR-14299292-015",
  "title": "Immediate Descendants of the Elliptic-Curve Groups of Maglione and Stanojkovski",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a prime p ≥ 5, an elliptic curve E: y² = x³ + ax + b over F_p and a point P ∈ E(F_p) other than the origin, Maglione and Stanojkovski defined a group G_{E,P}(F_p) of order p⁹, class 2 and exponent p. In the problem session of Oberwolfach Report 27/2025, Maglione asked which pairs (E, P) yield an immediate descendant, defined there as a central extension by Z/pZ of class 3 and order p¹⁰, and remarked that it would be remarkable if the sum, over the isomorphism classes of these groups, of the numbers of immediate descendants were a PORC function of p. We prove that G_{E,P}(F_p) has an immediate descendant if and only if P has order 2, and is terminal otherwise; for P of order 2 the nucleus has rank 2. For these pairs we give the number of immediate descendants of order p¹⁰, and of those of exponent p, as explicit functions of p, of |E[3](F_p)| and of the order of the stabilizer of P in the automorphism group of E; this contains the theorem of du Sautoy and Vaughan-Lee for y² = x³ − x and P = (0,0). Summed over the isomorphism classes, both numbers are PORC, with moduli 72 and 12: the dependence on |E[3](F_p)| cancels because the pairs (E, P) with given 3-torsion are parametrized by curves of genus 0. This answers both questions with the source's definition of an immediate descendant. If the descendants of order p¹¹, which are immediate descendants in O'Brien's sense, are counted as well, the PORC question remains open. The question is Maglione's; the groups, their classification and their automorphism groups are due to Maglione and Stanojkovski. This is an unrefereed note.",
  "result_type": "FULL_RESULT_ORDER_P10",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "keywords": [
    "p-groups of class two",
    "elliptic curves",
    "immediate descendants",
    "p-group generation",
    "nucleus",
    "p-covering group",
    "Lazard correspondence",
    "PORC",
    "Higman's PORC conjecture",
    "Tate normal form",
    "3-torsion",
    "Maglione",
    "Stanojkovski",
    "du Sautoy",
    "Vaughan-Lee",
    "Oberwolfach Report 27/2025 problems",
    "UnsolvedMath",
    "OWR-14299292-015",
    "math.GR",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-28",
  "publication_date": "2026-10-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299292-015/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299292-015/paper.pdf?v=2368fc5af8ff",
  "doi": "10.5281/zenodo.23709710",
  "zenodo_record_url": "https://zenodo.org/records/23709710",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Both questions are answered with the source's definition of an immediate descendant (order p^10); if the descendants of order p^11 are counted as well, the PORC question stays open; unrefereed.",
  "files": {
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      "sha256": "2368fc5af8fffe6a3dd605777f494a41a8bcaa00a740c46bd3cfa81372f3f5b6"
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    "source.zip": {
      "sha256": "3b1f3d0ff8549274438f30d8bcca26f64ed142a8655ad502a4de3e56184ac5ca"
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    "verification_report.md": {
      "sha256": "90a8198e87fef8a9e695de8bdab74f8eeba07461ba4e11b6736e1111426a4a8b"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}