{
  "id": "1305.0222",
  "version": "v3",
  "published": "2013-05-01T16:47:36.000Z",
  "updated": "2014-05-03T23:34:23.000Z",
  "title": "The Arithmetic of Curves Defined by Iteration",
  "authors": [
    "Wade Hindes"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-03T23:34:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic",
      "galois groups",
      "rational points",
      "quadratic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.0222H"
    }
  }
}