{
  "id": "1807.01937",
  "version": "v1",
  "published": "2018-07-05T10:43:26.000Z",
  "updated": "2018-07-05T10:43:26.000Z",
  "title": "Rational pullbacks of Galois covers",
  "authors": [
    "Pierre Dèbes",
    "Joachim Koenig",
    "François Legrand",
    "Danny Neftin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The finite subgroups of ${\\rm{PGL}}_2(\\mathbb{C})$ are shown to be the only finite groups $G$ with the following property: for some bound $r_0$, all Galois covers $X\\rightarrow \\mathbb{P}^1_\\mathbb{C}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along all non-constant rational maps $\\mathbb{P}^1_\\mathbb{C} \\rightarrow \\mathbb{P}^1_\\mathbb{C}$. For $G\\subset {\\rm PGL}_2(\\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A worthwhile consequence of the converse for inverse Galois theory is that, for $G\\not \\subset {\\rm PGL}_2(\\mathbb{C})$, letting the branch point number grow always provides truly new realizations $F/\\mathbb{C}(T)$ of $G$. Our approach also leads to some improvements of results of Buhler-Reichstein about generic polynomials with one parameter. For example, if $G$ is neither cyclic nor odd dihedral, no polynomial $P \\in \\mathbb{C}[T,Y]$ of Galois group $G$ can {parametrize}, via specialization of $T$, all Galois extensions $E/k$ of group $G$; here $k$ is some specific base field, not depending on $G$, viz. $k=\\mathbb{C}((V))(U)$. A final application, related to an old problem of Schinzel, provides, subject to the Birch and Swinnerton-Dyer conjecture, a $1$-parameter family of affine curves over some number field, all with a rational point, but with no rational generic point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-05T10:43:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galois covers",
      "rational pullbacks",
      "branch point number grow",
      "non-constant rational maps",
      "rational generic point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}