{
  "abstract": "Let $C \\subset \\Pbb^2_k$ be a smooth projective plane curve of degree $d \\ge 5$ over a number field $k$. We prove that any cubic arithmetic or geometric progression sequence on $C$ is finite. The proof rests on a prime-$p$ tower of cyclic Kummer covers controlled by the divisor of a linear-form quotient on $C$, combined with the Kadets--Vogt classification of curves with infinitely many cubic points, the Abramovich--Harris gonality descent, a Castelnuovo--Severi factorization for the elliptic maps supplied by Kadets--Vogt independently at consecutive levels of the tower, and a deck-transformation obstruction. We also obtain a conditional quartic result under a cyclotomic non-splitting hypothesis. The case $d = 4$ with $C(k) \\neq \\emptyset$ admits infinite cubic progressions, so no analogue of the main theorem can hold there.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04312",
  "author_total": 1,
  "authors": [
    {
      "name": "Eslam Badr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4q09qqj10fv7z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04312",
  "fulltext_url": null,
  "id": "4q09qqj10fv7z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04312",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220674967",
  "pdf_url": "https://arxiv.org/pdf/2610.04312",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4q09qqj10fv7z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04312",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04312"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cubic Progressions on Smooth Plane Curves",
  "title_text": "Cubic Progressions on Smooth Plane Curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/4q09qqj10fv7z",
  "venue": "arXiv",
  "versions": []
}