Kab has the Bogomolov property for canonical heights
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, Kab has the strong Bogomolov property for the canonical height of f. Combined with former results of Amoroso-Zannier, Pottmeyer and Ratazzi, this implies that Kab 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.
Related
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BibTeX
@misc{ferraguti2026bogomolov,
title = {{$\Kab$ has the Bogomolov property for canonical heights}},
author = {Andrea Ferraguti and Carlo Pagano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24938},
eprint = {2609.24938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24938},
}
RIS
TY - UNPB
TI - Kab has the Bogomolov property for canonical heights
AU - Andrea Ferraguti
AU - Carlo Pagano
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24938
UR - https://arxiv.org/abs/2609.24938
AB - 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^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^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.
ER -