{
  "id": "2408.08175",
  "version": "v1",
  "published": "2024-08-15T14:23:14.000Z",
  "updated": "2024-08-15T14:23:14.000Z",
  "title": "The geometric fundamental group of the affine line over a finite field",
  "authors": [
    "Henrik Russell"
  ],
  "comment": "33 pages",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \\'etale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a projective variety X a (non-commutative) universal affine pro-algebraic group Lu(X), such that for any given affine subvariety U of X any finite and \\'etale Galois covering of U over F is a pull-back of a Galois covering of a quotient Lu(X,U) of Lu(X). Then the geometric fundamental group of U is a completion of the k-points of Lu(X,U), where k is an algebraic closure of F. We obtain explicit descriptions of the universal affine groups Lu(X,U) for U the affine line and the punctured affine line over F.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-15T14:23:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H30",
      "14L17",
      "14G15"
    ],
    "keywords": [
      "geometric fundamental group",
      "finite field",
      "punctured affine line",
      "universal affine pro-algebraic group lu",
      "geometric etale fundamental groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}