Quantitative Equidistribution of Small Points for Canonical Heights
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914996737277952 |
|---|---|
| author | Yap, Jit Wu |
| author_facet | Yap, Jit Wu |
| contents | Let $X$ be a smooth projective variety defined over a number field $K$ and let $φ: X \to X$ a polarized endomorphism of degree $d \geq 2$. Let $\widehat{h}_φ$ be the canonical height associated to $φ$ on $X(\overline{K})$. Given a generic sequence of points $(x_n)$ with $\widehat{h}_φ(x_n) \to 0$ and a place $v \in M_K$, Yuan [Yua08] has shown that the conjugates of $x_n$ equidistribute to the canonical measure $μ_{φ,v}$. When $v$ is archimedean, we will prove a quantitative version of Yuan's result. We give two applications of our result to polarized endomorphisms $φ$ 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 $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative Equidistribution of Small Points for Canonical Heights Yap, Jit Wu Number Theory Dynamical Systems Let $X$ be a smooth projective variety defined over a number field $K$ and let $φ: X \to X$ a polarized endomorphism of degree $d \geq 2$. Let $\widehat{h}_φ$ be the canonical height associated to $φ$ on $X(\overline{K})$. Given a generic sequence of points $(x_n)$ with $\widehat{h}_φ(x_n) \to 0$ and a place $v \in M_K$, Yuan [Yua08] has shown that the conjugates of $x_n$ equidistribute to the canonical measure $μ_{φ,v}$. When $v$ is archimedean, we will prove a quantitative version of Yuan's result. We give two applications of our result to polarized endomorphisms $φ$ 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 $X$. |
| title | Quantitative Equidistribution of Small Points for Canonical Heights |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2410.21679 |