{
  "id": "2106.13003",
  "version": "v1",
  "published": "2021-06-24T13:31:43.000Z",
  "updated": "2021-06-24T13:31:43.000Z",
  "title": "A Bogomolov property for the canonical height of maps with superattracting periodic points",
  "authors": [
    "Nicole R. Looper"
  ],
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "We prove that if $f$ is a polynomial over a number field $K$ with a finite superattracting periodic point and a non-archimedean place of bad reduction, then there is an $\\epsilon>0$ such that only finitely many $P\\in K^{\\text{ab}}$ have canonical height less than $\\epsilon$ with respect to $f$. The key ingredient is the geometry of the filled Julia set at a place of bad reduction. We also prove a conditional uniform boundedness result for the $K$-rational preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in $K$. We further prove unconditional analogues of these results in the function field setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-24T13:31:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G50",
      "37P05",
      "37P35",
      "37P40"
    ],
    "keywords": [
      "canonical height",
      "bogomolov property",
      "conditional uniform boundedness result",
      "bad reduction",
      "finite superattracting periodic point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}