{
  "id": "2310.03543",
  "version": "v1",
  "published": "2023-10-05T13:53:45.000Z",
  "updated": "2023-10-05T13:53:45.000Z",
  "title": "Stability of 2-class groups in $\\mathbb{Z}_2$-extension of certain real quadratic number fields",
  "authors": [
    "H Laxmi",
    "Anupam Saikia"
  ],
  "comment": "14 pages, 3 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a real quadratic field $K= \\mathbb{Q}(\\sqrt{d})$ with $d$ having three distinct prime factors, it has been proven that under certain conditions, the Iwasawa module $X_{\\infty}$ corresponding to the cyclotomic $\\mathbb{Z}_2$-extension of $K$ is cyclic. In this paper, with some elementary assumptions on the prime factors of $d$, we show that $X_{\\infty}$ is isomorphic to $\\mathbb{Z}/2\\mathbb{Z}$. Consequently, we prove that the Iwasawa $\\lambda$-invariant for such fields is equal to 0, validating Greenberg's conjecture for these fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-05T13:53:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11R11",
      "11R23"
    ],
    "keywords": [
      "real quadratic number fields",
      "distinct prime factors",
      "real quadratic field",
      "validating greenbergs conjecture",
      "elementary assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}