{
  "id": "2201.04291",
  "version": "v1",
  "published": "2022-01-12T03:55:23.000Z",
  "updated": "2022-01-12T03:55:23.000Z",
  "title": "Congruences for odd class numbers of quadratic fields with odd discriminant",
  "authors": [
    "Jigu Kim",
    "Yoshinori Mizuno"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For any distinct two primes $p_1\\equiv p_2\\equiv 3$ $(\\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{-p_1})$, $\\mathbb{Q}(\\sqrt{-p_2})$ and $\\mathbb{Q}(\\sqrt{p_1p_2})$, respectively. Let $\\omega_{p_1p_2}:=\\frac{1+\\sqrt{p_1p_2}}{2}$ and let $\\Psi(\\omega_{p_1p_2})$ be the Hirzebruch sum of $\\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\\equiv h(p_1p_2)\\Psi(\\omega_{p_1p_2})/n$ $(\\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\\min\\{p_1,p_2\\}>3$ (respectively, otherwise). We also consider the real quadratic order with conductor $2$ in $\\mathbb{Q}(\\sqrt{p_1p_2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-12T03:55:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11A55",
      "11F20"
    ],
    "keywords": [
      "odd class numbers",
      "quadratic fields",
      "odd discriminant",
      "congruences",
      "real quadratic order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}