Quantitative Equidistribution of Small Points for Canonical Heights

TL;DR

This paper provides a quantitative analysis of how small points distribute on varieties with respect to canonical heights, offering explicit convergence rates and bounds for periodic points and points of bounded degree.

Contribution

It introduces a quantitative version of Yuan's equidistribution theorem for archimedean places and applies it to derive exponential convergence rates and degree bounds for periodic points on surfaces and abelian varieties.

Findings

Exponential rate of convergence for periodic points to the equilibrium measure.

Exponential lower bounds on the degree of fields containing periodic points.

Upper bounds on degrees of points with small Neron--Tate height on abelian varieties.

Abstract

Let $X$ be a smooth projective variety defined over a number field $K$ and let $\varphi: X \to X$ a polarized endomorphism of degree $d \geq 2$. Let $\widehat{h}_{\varphi}$ be the canonical height associated to $\varphi$ on $X(\overline{K})$. Given a generic sequence of points $(x_n)$ with $\widehat{h}_{\varphi}(x_n) \to 0$ and a place $v \in M_K$, Yuan [Yua08] has shown that the conjugates of $x_n$ equidistribute to the canonical measure $\mu_{\varphi,v}$. When $v$ is archimedean, we will prove a quantitative version of Yuan's result. We give two applications of our result to polarized endomorphisms $\varphi$ of smooth projective surfaces that are defined over a number field $K$. The first is an exponential rate of convergence for periodic points of period $n$ to the equilibrium measure and the second is an exponential lower bound on the degree of the extension containing all periodic points of period $n$. When $X$ is an abelian variety, we also give an upper bound on the smallest degree of a hypersurface that contains all points $x \in X(\overline{K})$ satisfying $[K(x):K] \leq D$ and $\widehat{h}_X(x) \leq \frac{c}{D^8}$ for some fixed constant $c > 0$ where $\widehat{h}_X$ is the Neron--Tate height for $X$.