{
  "id": "1303.2248",
  "version": "v1",
  "published": "2013-03-09T18:21:44.000Z",
  "updated": "2013-03-09T18:21:44.000Z",
  "title": "Faithful actions of the absolute Galois group on connected components of moduli spaces",
  "authors": [
    "Ingrid Bauer",
    "Fabrizio Catanese",
    "Fritz Grunewald"
  ],
  "comment": "24 pages, extends and corrects a previous article arXiv:0706.1466",
  "categories": [
    "math.AG"
  ],
  "abstract": "We give a canonical procedure associating to an algebraic number a first a hyperelliptic curve C_a, and then a triangle curve (D_a, G_a) obtained through the normal closure of an associated Belyi function. In this way we show that the absolute Galois group Gal(\\bar{\\Q} /\\Q) acts faithfully on the set of isomorphism classes of marked triangle curves, and on the set of connected components of marked moduli spaces of surfaces isogenous to a higher product (these are the free quotients of a product C_1 x C_2 of curves of respective genera g_1, g_2 >= 2 by the action of a finite group G). We show then, using again the surfaces isogenous to a product, first that it acts faithfully on the set of connected components of moduli spaces of surfaces of general type (amending an incorrect proof in a previous ArXiv version of the paper); and then, as a consequence, we obtain that for every element \\sigma \\in \\Gal(\\bar{\\Q} /\\Q), not in the conjugacy class of complex conjugation, there exists a surface of general type X such that X and the Galois conjugate surface X^{\\sigma} have nonisomorphic fundamental groups. Using polynomials with only two critical values, we can moreover exhibit infinitely many explicit examples of such a situation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-09T18:21:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R32",
      "14J10",
      "14J29",
      "M99"
    ],
    "keywords": [
      "absolute galois group",
      "moduli spaces",
      "connected components",
      "faithful actions",
      "triangle curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2248B"
    }
  }
}