{
  "abstract": "We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted $p$-ramification. Let $K$ be an algebraic number field, and $p$ a prime number which splits into two distinct primes $\\mathfrak{p}$, $\\mathfrak{p}'$ in $K$. Assume that $K$ and $p$ satisfies several (somewhat strict) conditions. Let $K_\\infty /K$ be the cyclotomic $\\mathbb{Z}_p$-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module $X (K_\\infty)$ by using abelian $p$-extensions unramified outside $\\mathfrak{p}$. We give a sufficient condition for $|X (K_\\infty)|$ to be finite in terms of such extensions. We also give a similar criterion for $X (K_\\infty)$ to be finitely generated over $\\mathbb{Z}_p$. In the latter part of the present paper, we consider the case where $k = \\mathbb{Q} (\\sqrt{q})$ with an odd prime number $q$ and apply our results to the cyclotomic $\\mathbb{Z}_2$-extension $k_\\infty /k$. We give a necessary and sufficient condition for $X (k_\\infty)$ to be cyclic over $\\mathbb{Z}_2$, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25866",
  "author_total": 2,
  "authors": [
    {
      "name": "Tsuyoshi Itoh"
    },
    {
      "name": "Naoki Kumakawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0nhw32b8znbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25866",
  "fulltext_url": null,
  "id": "c0nhw32b8znbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25866",
  "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": "W7214116401",
  "pdf_url": "https://arxiv.org/pdf/2609.25866",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0nhw32b8znbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.25866",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25866"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Abelian $p$-extensions with restricted $p$-ramification and the cyclotomic $\\mathbb{Z}_2$-extension of $\\mathbb{Q} (\\sqrt{q})$",
  "title_text": "Abelian p-extensions with restricted p-ramification and the cyclotomic ℤ_2-extension of ℚ (√q)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/c0nhw32b8znbh",
  "venue": "arXiv",
  "versions": []
}