On the Number of Small Points for Rational Maps

TL;DR

The paper establishes a uniform Baker-type bound for small points of rational maps of degree $d$ with $s$ places of bad reduction, showing that the number of points with canonical height below $(c_1/s)h_{\operatorname{rat}_d}(\langle f\rangle)$ is at most $c_2 s\log s$, with constants depending only on $d$. The authors develop a robust framework linking local Arakelov--Green functions to a global moduli height via the moduli space $\operatorname{rat}_d$ and a minimal resultant height, and they extend the dynamics to degenerations using Berkovich spaces over Banach rings, following Luo’s ultrafilter approach formalized by Favre--Gong. By constructing uniform, degeneration-based lower bounds for the dynamical Arakelov--Green function across archimedean and non-archimedean places, they generalize Baker’s results from polynomials to rational maps and obtain corollaries such as uniform bounds on preperiodic points and lower height gaps for non-preperiodic points, with analogous results over function fields. The methodology provides a bridge between complex and non-archimedean dynamics via Arakelov theory and Berkovich potential theory, yielding practically significant bounds that are uniform across families of rational maps.

Abstract

Let $K$ be a number field and $f: \mathbb{P}^1 \to \mathbb{P}^1$ a rational map of degree $d \geq 2$ with at most $s$ places of bad reduction, where we include all archimedean places. We prove that there exists constants $c_1,c_2 > 0$, depending only on $d$ and not on $f$ or $K$, such that $$ \# \left\{ x \in \mathbb{P}^1(K) \mid \widehat{h}_f(x) \leq \frac{c_1}{s} h_{\operatorname{rat}_d}(\langle f \rangle) \right\} \leq c_2 s \log(s). $$ Here, $\operatorname{rat}_d$ is the moduli space of rational maps up to conjugacy, $h_{\operatorname{rat}_d}$ is an ample height and $\langle f \rangle$ is the equivalence class associated to $f$. This gives a uniform version of a theorem of Baker as well as generalizing the results of Benedetto and Looper from polynomials to rational maps. The main tool used is the degeneration of sequences of rational maps by Luo which has been recently formalized by Favre-Gong via Berkovich spaces.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Definition 3.1
• Proposition 3.1
• proof