Quantitative Equidistribution of Small Points for Canonical Heights

Let $X$ be a smooth projective variety defined over a number field $K$ and let $\phi :X\to X$ a polarized endomorphism of degree $d\ge 2$. Let ${\widehat{h}}_{\phi }$ be the canonical height associated to $\phi$ on $X(\bar{K})$. Given a generic sequence of points $(x_n)$ with ${\widehat{h}}_{\phi }(x_n)\to 0$ and a place $v\in M_K$, Yuan [Yua08] has shown that the conjugates of $x_n$ equidistribute to the canonical measure ${\mu }_{\phi ,v}$. When $v$ is archimedean, we will prove a quantitative version of Yuan’s result. We give two applications of our result to polarized endomorphisms $\phi$ 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 $x\in X(\bar{K})$ satisfying $[K(x):K]\le D$ and ${\widehat{h}}_X(x)\le \frac{c}{D^8}$ for some fixed constant $c>0$ where ${\widehat{h}}_X$ is the Neron–Tate height for $X$.