{
  "id": "1007.2318",
  "version": "v1",
  "published": "2010-07-14T12:52:38.000Z",
  "updated": "2010-07-14T12:52:38.000Z",
  "title": "On some arithmetic properties of Siegel functions (II)",
  "authors": [
    "Ho Yun Jung",
    "Ja Kyung Koo",
    "Dong Hwa Shin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K$ be an imaginary quadratic field with discriminant $d_K\\leq-7$. We deal with problems of constructing normal bases between abelian extensions of $K$ by making use of singular values of Siegel functions. First, we show that a criterion achieved from the Frobenius determinant relation enables us to find normal bases of ring class fields of orders of bounded conductors depending on $d_K$ over $K$. Next, denoting by $K_{(N)}$ the ray class field modulo $N$ of $K$ for an integer $N\\geq2$ we consider the field extension $K_{(p^2m)}/K_{(pm)}$ for a prime $p\\geq5$ and an integer $m\\geq1$ relatively prime to $p$ and then find normal bases of all intermediate fields over $K_{(pm)}$ by utilizing Kawamoto's arguments. And, we further investigate certain Galois module structure of the field extension $K_{(p^{n}m)}/K_{(p^{\\ell}m)}$ with $n\\geq 2\\ell$, which would be an extension of Komatsu's work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-07-14T12:52:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11F20",
      "11R37",
      "11Y40"
    ],
    "keywords": [
      "siegel functions",
      "arithmetic properties",
      "normal bases",
      "frobenius determinant relation enables",
      "ray class field modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.2318J"
    }
  }
}