{
  "abstract": "For the quadratic family $f_c(x) = x^2 + c$ over $\\Q$, we isolate a general $S$-unit and ABC template for rational periodic cycles: under the ABC conjecture, any rational quantity built from the consecutive-point sums of a rational period-$n$ cycle whose value and shift-by-$1$ are both $S$-units (with $S$ the support of the denominator of $c$) has logarithmic height at most $(1+\\eps)\\sum_{p \\in S} \\log p + O_\\eps(1)$. Specializing to period $3$ in the change of coordinates $u = x_0 + x_1$ and $v = x_0 + x_2$, the period-$3$ condition reduces to $v = -1/(1+u)$, forcing the parameter $t = u$ to satisfy $t, t+1 \\in \\Z[S^{-1}]^\\times$; the template then yields, under ABC, $\\max\\{|a|,|b|,|a+b|\\} \\le C_\\eps \\rad(d)^{1+\\eps}$ for $t = a/b$ in lowest terms. We also characterize the period-$3$ locus on the $(u,v)$-plane intrinsically, isolating the ghost curve. Extending the height bound to higher periods remains an open structural question.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39820",
  "author_total": 2,
  "authors": [
    {
      "name": "Antti Käenmäki"
    },
    {
      "name": "Pekka Kosunen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2t7tftytrmh6y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39820",
  "fulltext_url": null,
  "id": "2t7tftytrmh6y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39820",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39820",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2t7tftytrmh6y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39820",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39820"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Rational points of exact period three for the quadratic family",
  "title_text": "Rational points of exact period three for the quadratic family",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/2t7tftytrmh6y",
  "venue": "arXiv",
  "versions": []
}