{
  "id": "2012.01202",
  "version": "v1",
  "published": "2020-12-02T13:38:17.000Z",
  "updated": "2020-12-02T13:38:17.000Z",
  "title": "Simultaneous indivisibility of class numbers of pairs of real quadratic fields",
  "authors": [
    "Jaitra Chattopadhyay",
    "Anupam Saikia"
  ],
  "comment": "8 pages. Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a square-free integer $t$, Byeon \\cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\\mathbb{Q}(\\sqrt{D})$ and $\\mathbb{Q}(\\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible by $3$. In the same spirit, we prove that for a given integer $t \\geq 1$ with $t \\equiv 0 \\pmod {4}$, a positive proportion of fundamental discriminants $D > 0$ exist for which the class numbers of both the real quadratic fields $\\mathbb{Q}(\\sqrt{D})$ and $\\mathbb{Q}(\\sqrt{D + t})$ are indivisible by $3$. This also addresses the complement of a weak form of a conjecture of Iizuka in \\cite{iizuka}. As an application of our main result, we obtain that for any integer $t \\geq 1$ with $t \\equiv 0 \\pmod{12}$, there are infinitely many pairs of real quadratic fields $\\mathbb{Q}(\\sqrt{D})$ and $\\mathbb{Q}(\\sqrt{D + t})$ such that the Iwasawa $\\lambda$-invariants associated with the basic $\\mathbb{Z}_{3}$-extensions of both $\\mathbb{Q}(\\sqrt{D})$ and $\\mathbb{Q}(\\sqrt{D + t})$ are $0$. For $p = 3$, this supports Greenberg's conjecture which asserts that $\\lambda_{p}(K) = 0$ for any prime number $p$ and any totally real number field $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-02T13:38:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real quadratic fields",
      "class numbers",
      "simultaneous indivisibility",
      "totally real number field",
      "supports greenbergs conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}