{
  "id": "2210.02668",
  "version": "v1",
  "published": "2022-10-06T04:08:58.000Z",
  "updated": "2022-10-06T04:08:58.000Z",
  "title": "Congruences on the class numbers of $\\mathbb{Q}(\\sqrt{\\pm 2p})$ for $p\\equiv3$ $(\\text{mod }4)$ a prime",
  "authors": [
    "Jigu Kim",
    "Yoshinori Mizuno"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a prime $p\\equiv 3$ $(\\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\\mathbb{Q}(\\sqrt{-2p})$ and $\\mathbb{Q}(\\sqrt{2p})$, respectively. Let $\\Psi(\\xi)$ be the Hirzebruch sum of a quadratic irrational $\\xi$. We show that $h(-8p)\\equiv h(8p)\\Big(\\Psi(2\\sqrt{2p})/3-\\Psi\\big((1+\\sqrt{2p})/2\\big)/3\\Big)$ $(\\text{mod }16)$. Also, we show that $h(-8p)\\equiv 2h(8p)\\Psi(2\\sqrt{2p})/3$ $(\\text{mod }8)$ if $p\\equiv 3$ $(\\text{mod }8)$, and $h(-8p)\\equiv \\big(2h(8p)\\Psi(2\\sqrt{2p})/3\\big)+4$ $(\\text{mod }8)$ if $p\\equiv 7$ $(\\text{mod }8)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-06T04:08:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11A55",
      "11F20"
    ],
    "keywords": [
      "class numbers",
      "congruences",
      "hirzebruch sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}