{
  "id": "2410.21679",
  "version": "v1",
  "published": "2024-10-29T03:00:47.000Z",
  "updated": "2024-10-29T03:00:47.000Z",
  "title": "Quantitative Equidistribution of Small Points for Canonical Heights",
  "authors": [
    "Jit Wu Yap"
  ],
  "comment": "Comments welcome!",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "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 $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-29T03:00:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "canonical height",
      "small points",
      "quantitative equidistribution",
      "number field",
      "periodic points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}