{
  "id": "1710.02257",
  "version": "v1",
  "published": "2017-10-06T02:14:44.000Z",
  "updated": "2017-10-06T02:14:44.000Z",
  "title": "Finite index theorems for iterated Galois groups of cubic polynomials",
  "authors": [
    "Andrew Bridy",
    "Thomas J. Tucker"
  ],
  "comment": "36 pages, 4 figures",
  "categories": [
    "math.NT",
    "math.DS"
  ],
  "abstract": "Let $K$ be a number field or a function field. Let $f\\in K(x)$ be a rational function of degree $d\\geq 2$, and let $\\beta\\in\\mathbb{P}^1(K)$. For all $n\\in\\mathbb{N}\\cup\\{\\infty\\}$, the Galois groups $G_n(\\beta)=\\text{Gal}(K(f^{-n}(\\beta))/K)$ embed into $\\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\\text{Aut}(T_\\infty):G_\\infty]<\\infty$. When $f$ is a cubic polynomial and $K$ is a function field of transcendence degree $1$ over an algebraic extension of $\\mathbb{Q}$, we resolve this problem by proving a list of necessary and sufficient conditions for finite index. This is the first result that gives necessary and sufficient conditions for finite index, and can be seen as a dynamical analog of the Serre Open Image Theorem. When $K$ is a number field, our proof is conditional on both the $abc$ conjecture for $K$ and Vojta's conjecture for blowups of $\\mathbb{P}^1\\times\\mathbb{P}^1$. We also use our approach to solve some natural variants of the finite index problem for modified trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-06T02:14:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R32",
      "37P05",
      "37P15",
      "37P30",
      "11G50",
      "14G25"
    ],
    "keywords": [
      "finite index theorems",
      "iterated galois groups",
      "cubic polynomial",
      "sufficient conditions",
      "function field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}