A Bogomolov property for the canonical height of maps with superattracting periodic points

TL;DR

This paper establishes a finiteness property for algebraic points with small canonical height under certain polynomial maps with superattracting periodic points, especially at places of bad reduction, and explores related uniform bounds.

Contribution

It proves a Bogomolov-type property for canonical heights of polynomials with superattracting points, including new results at places of bad reduction and in the function field setting.

Findings

Finiteness of points with small canonical height over number fields.

Conditional uniform boundedness of preperiodic points.

Unconditional results in the function field case.

Abstract

We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $\epsilon>0$ such that only finitely many $P\in K^{\text{ab}}$ have canonical height less than $\epsilon$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove unconditional analogues of these results in the function field setting.