{
  "abstract": "Let $p$ be an odd prime and let $K_\\infty/K$ be a $\\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $μ$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $δ>0$ such that $d_n=δp^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\\geq2$, as $n\\to\\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22297",
  "author_total": 1,
  "authors": [
    {
      "name": "Anwesh Ray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0cehv5agn159t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22297",
  "fulltext_url": null,
  "id": "0cehv5agn159t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22297",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214000015",
  "pdf_url": "https://arxiv.org/pdf/2609.22297",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0cehv5agn159t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22297",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22297"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Iwasawa theory of class field towers",
  "title_text": "Iwasawa theory of class field towers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0cehv5agn159t",
  "venue": "arXiv",
  "versions": []
}