{
  "id": "1103.3853",
  "version": "v1",
  "published": "2011-03-20T14:11:58.000Z",
  "updated": "2011-03-20T14:11:58.000Z",
  "title": "On some notions of good reduction for endomorphisms of the projective line",
  "authors": [
    "Jung Kyu Canci",
    "Giulio Peruginelli",
    "Dajano Tossici"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\Phi$ be an endomorphism of $\\SR(\\bar{\\Q})$, the projective line over the algebraic closure of $\\Q$, of degree $\\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $\\Phi$ has critically good reduction at $v$ if any pair of distinct ramification points of $\\Phi$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $\\Phi$ has simple good reduction at $v$ if the map $\\Phi_v$, the reduction of $\\Phi$ modulo $v$, has the same degree of $\\Phi$. We prove that if $\\Phi$ has critically good reduction at $v$ and the reduction map $\\Phi_v$ is separable, then $\\Phi$ has simple good reduction at $v$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-20T14:11:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H25",
      "37P05",
      "37P35"
    ],
    "keywords": [
      "projective line",
      "endomorphism",
      "distinct ramification points",
      "algebraic closure",
      "branch points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3853C"
    }
  }
}