{
  "id": "2408.11916",
  "version": "v1",
  "published": "2024-08-21T18:06:36.000Z",
  "updated": "2024-08-21T18:06:36.000Z",
  "title": "Gerth's heuristics for a family of quadratic extensions of certain Galois number fields",
  "authors": [
    "C. G. K. Babu",
    "R. Bera",
    "J. Sivaraman",
    "B. Sury"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit $$ \\lim_{x \\to \\infty} \\frac{\\sum_{0 < D \\le X, \\atop{ \\text{squarefree} }} |{\\rm Cl}^2_{\\Q(\\sqrt{D})}/{\\rm Cl}^4_{\\Q(\\sqrt{D})}|^m}{\\sum_{0 < D \\le X, \\atop{ \\text{squarefree} }} 1} $$ exists and proposed a value for the limit. Gerth's conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $|{\\rm Cl}^2_{\\L}/{\\rm Cl}^4_{\\L}|^m$, where $\\L$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\\Z$ splits.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T18:06:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galois number field",
      "quadratic extensions",
      "gerths heuristics",
      "average value",
      "gerth generalised cohen-lenstra heuristics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}