{
  "id": "1208.5794",
  "version": "v2",
  "published": "2012-08-28T21:02:18.000Z",
  "updated": "2017-03-29T16:10:25.000Z",
  "title": "On quadratic rational maps with prescribed good reduction",
  "authors": [
    "Clayton Petsche",
    "Brian Stout"
  ],
  "comment": "This version fixes a few LaTeX errors",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\\phi:{\\mathbb P}^1\\to{\\mathbb P}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\\mathcal M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside $S$. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-28T21:02:18.000Z",
      "abstract": "Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\\phi:\\PP^1\\to\\PP^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space $\\Mcal_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside $S$. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-03-29T16:10:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37P45",
      "14G25",
      "37P15"
    ],
    "keywords": [
      "quadratic rational maps",
      "second main result establishes",
      "geometric shafarevich-type non-zariski-density result",
      "first main result",
      "moduli space"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5794P"
    }
  }
}