{
  "id": "2004.08860",
  "version": "v1",
  "published": "2020-04-19T14:18:15.000Z",
  "updated": "2020-04-19T14:18:15.000Z",
  "title": "On representations of $\\mathrm{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})$ and $\\mathrm{Aut}(\\hat{F}_d)$",
  "authors": [
    "Frauke M. Bleher",
    "Ted Chinburg",
    "Alexander Lubotzky"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "It is well known that $G=\\mathrm{Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q})$, the absolute Galois group of the field of rational numbers $\\mathbb{Q}$, is embedded as an (infinite index) subgroup in $A=\\mathrm{Aut}(\\hat{F}_2)$, the automorphism group of the free profinite group on 2 generators $\\hat{F}_2$. In this note, we prove a \"super-rigidity\" result for some classical Galois representations of $G$. Namely, let $X$ be a smooth projective irreducible curve over $\\overline{\\mathbb{Q}}$, realized as a cover $\\lambda:X\\to \\mathbb{P}^1_{\\overline{\\mathbb{Q}}}$ that is unramified outside $\\{0,1,\\infty\\}$, and let $\\ell$ is a prime number. Then the action of a natural finite index subgroup of $G$ on the $\\ell$-adic Tate module of the generalized Jacobian of $X$ with respect to the ramification locus of $\\lambda$, can be extended, up to a finite index subgroup, to an action of a finite index subgroup of $A$. Moreover, this holds even for the product of the Tate modules of the generalized Jacobian of $X$ over all primes $\\ell$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-19T14:18:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G32",
      "20F34",
      "14H30"
    ],
    "keywords": [
      "natural finite index subgroup",
      "absolute galois group",
      "free profinite group",
      "adic tate module",
      "generalized jacobian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}