{
  "id": "math/9908175",
  "version": "v2",
  "published": "1999-08-31T13:27:10.000Z",
  "updated": "1999-09-01T07:28:51.000Z",
  "title": "The 2-primary class group of certain hyperelliptic curves",
  "authors": [
    "Gunther Cornelissen"
  ],
  "comment": "10 pages, LaTeX, uses `a4'",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the Jacobian J(X) of X depends only on the parity of |W(X)|, but for large enough |F|, there exist two such curves X and X' with |W(X)|=|W(X')|, such that J(X) has (and J(X') does not have) a G-invariant 4-torsion point. The problem is equivalent to a study of the 2-,4- and 8-rank of the class number of the maximal order in the function field of such curves, and is investigated via the 2-primary class field tower. Contrary to the case of number fields, the ambiguous class depends on the discriminant, and a governing field for the 8-rank of such function fields is not known.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-09-01T07:28:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "14H40"
    ],
    "keywords": [
      "class group",
      "function field",
      "imaginary hyperelliptic curve",
      "class field tower",
      "weierstrass points"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......8175C"
    }
  }
}