{
  "abstract": "In this paper, we study the unramified Iwasawa module over the cyclotomic $\\mathbb{Z}_2$-extension of the real quadratic field $\\mathbb{Q}(\\sqrt{p_1p_2p_3p_4})$, where $p_1, p_2, p_3$, and $p_4$ are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type $\\mathbb{Z}/4\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z}$ and $2$-class group of type $\\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z} \\oplus \\mathbb{Z}/2\\mathbb{Z}$. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic $\\mathbb{Z}_2$-extension is totally ramified and satisfies Greenberg's conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23063",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuito Saito"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kazq93z962bw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23063",
  "fulltext_url": null,
  "id": "5kazq93z962bw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23063",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7213995927",
  "pdf_url": "https://arxiv.org/pdf/2609.23063",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kazq93z962bw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23063",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23063"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On $\\mathbb{Z}_2$-extensions of real quadratic fields with class group of $2$-rank three",
  "title_text": "On ℤ_2-extensions of real quadratic fields with class group of $2$-rank three",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5kazq93z962bw",
  "venue": "arXiv",
  "versions": []
}