{
  "id": "2205.07349",
  "version": "v1",
  "published": "2022-05-15T18:20:36.000Z",
  "updated": "2022-05-15T18:20:36.000Z",
  "title": "Moduli spaces of quadratic maps: arithmetic and geometry",
  "authors": [
    "Rohini Ramadas"
  ],
  "comment": "6 pages, 2 figures, comments welcome",
  "categories": [
    "math.DS",
    "math.AG",
    "math.NT"
  ],
  "abstract": "We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\\in\\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\\mathrm{Per}_n(0)$ is the $1$-dimensional moduli space of quadratic rational maps on $\\mathbb{P}^1$ with an $n$-periodic critical point. We show that if $G_n$ is irreducible over $\\mathbb{Q}$, then $\\mathrm{Per}_n(0)$ is irreducible over $\\mathbb{C}$. To do this, we exhibit a $\\mathbb{Q}$-rational smooth point on a projective completion of $\\mathrm{Per}_n(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\\mathrm{Per}_n(0)$ itself has no $\\mathbb{Q}$-rational points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-15T18:20:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quadratic maps",
      "dimensional moduli space",
      "periodic critical point",
      "uniform boundedness conjecture",
      "th gleason polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}