{
  "abstract": "Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\\varnothing}(K)(p)$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\\mathrm{Cl}(K)[p]$ to $\\mathrm{Cl}(K)/p$; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-$p$ extension attached to $x$. These maps determine all triple Massey products on $H^1(G,\\mathbb F_p)$, and with them the cubic initial relations of $G$. Suppose that the $p$-class rank $d=\\dim_{\\mathbb F_p}\\mathrm{Cl}(K)/p$ is at least three. The $(d-1)\\times(d-1)$ minors of the family $x\\mapsto D_x$ define a subscheme $Σ_D$ of $\\mathbb P^{d-1}_{\\mathbb F_p}$, an invariant of $K$, the norm-degeneracy scheme. We prove: if the rank condition $\\mathrm{rk}\\,D_x=d-2$ holds transversally at a point of $Σ_D$, over some finite extension of $\\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $Σ_D$ is smooth of dimension $d-3$ at the point; at $p=3$ the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of $p$-class rank at least three with $|D_K|<2^{30}$. Only $p=3$, 5, and 7 occur. The criterion decides 206 of the 207 fields at $p=5$ and 7, and 505 of the 12 750 fields at $p=3$, where the Bockstein condition restricts its reach. A direct computation with the cubic initial relations settles the remaining field at $p=5$ and a further 11 765 at $p=3$, 26 of them not mild, while 480 remain undecided. In all, the $p$-class tower group is proved mild for 12 451 of the 12 957 fields. These appear to be the first number fields for which the full maximal everywhere unramified pro-$p$ Galois group is proved to be mild, and hence of cohomological dimension 2.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.17072",
  "author_total": 1,
  "authors": [
    {
      "name": "Denis Vogel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94xazv327g3yk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.17072",
  "fulltext_url": null,
  "id": "94xazv327g3yk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.17072",
  "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": "W7203794998",
  "pdf_url": "https://arxiv.org/pdf/2608.17072",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94xazv327g3yk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.17072",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.17072"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Mild p-Class Tower Groups of Imaginary Quadratic Fields",
  "title_text": "Mild p-Class Tower Groups of Imaginary Quadratic Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/94xazv327g3yk",
  "venue": "arXiv",
  "versions": []
}