{
  "id": "1904.07743",
  "version": "v1",
  "published": "2019-04-16T14:53:13.000Z",
  "updated": "2019-04-16T14:53:13.000Z",
  "title": "p-adic equidistribution of CM points",
  "authors": [
    "Daniel Disegni"
  ],
  "comment": "21 pages, comments welcome",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. We study the equidistribution problem in the Berkovich analytification $X_{p}^{\\rm an}$ of $X_{{\\bf Q}_{p}}$. We partition the set of CM points of sufficiently high conductor in $X_{{\\bf Q}_{p}}$ into finitely many \\emph{basins} $B_{V}$, indexed by the irreducible components $V $ of the mod-$p$ reduction of the canonical model of $X$. We prove that a sequence $z_{n}$ of local Galois orbits of CM points with $p$-conductor going to infinity has a limit in $X_{p}^{\\rm an}$ if and only if it is eventually supported in a single basin $B_{V}$. If so, the limit is the unique point of $X_{p}^{\\rm an}$ whose mod-$p$ reduction is the generic point of $V$. The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasicanonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin--Tate space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-16T14:53:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G15",
      "14K22"
    ],
    "keywords": [
      "cm points",
      "p-adic equidistribution",
      "local galois orbits",
      "equidistribution properties",
      "modulo primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}