On the Number of Small Points for 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.