{
  "abstract": "We show that for any number field $K$ and any rational map $f$ in $K(x)$ of degree at least $2$ that is not conjugate to a power, (signed) Chebyshev or CM Lattès map, $K^{\\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. Combined with former results of Amoroso-Zannier, Pottmeyer and Ratazzi, this implies that $K^{\\mathrm{ab}}$ has the Bogomolov property for canonical heights of any rational map of degree at least $2$. We also classify the pairs $(f,α)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews--Petsche conjecture in the greater generality of rational maps over a number field and infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with \\emph{Astra}. The key insight consists of applying Yuan's equidistribution results, followed by a classification of $(f,f)$-preperiodic curves due to Pakovich and Beaumont followed by an additional application of equidistribution on a parametrized preperiodic curve.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24938",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrea Ferraguti"
    },
    {
      "name": "Carlo Pagano"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cdr8qt16hq4rt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24938",
  "fulltext_url": null,
  "id": "cdr8qt16hq4rt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24938",
  "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": "W7214029770",
  "pdf_url": "https://arxiv.org/pdf/2609.24938",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cdr8qt16hq4rt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24938",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24938"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$\\Kab$ has the Bogomolov property for canonical heights",
  "title_text": "Kab has the Bogomolov property for canonical heights",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/cdr8qt16hq4rt",
  "venue": "arXiv",
  "versions": []
}