{
  "abstract": "For an odd prime $p$, let $A_j$ be the $ω^j$-eigenspace of the $p$-primary class group of $\\mathbb Q(ζ_p)$. Fix an even integer $d\\ge4$, put $N=(p-1)/d$, and let $U_d=(\\mathbb Z/d\\mathbb Z)^\\times$. For a relative density-one set of primes $p\\equiv d+1\\pmod{2d}$, we prove that the odd block $\\bigoplus_{a\\in U_d}A_{aN}$ has order at most $p^{φ(d)/2-1}$, and reflection shows that the even block $\\bigoplus_{a\\in U_d}A_{p-aN}$ has $p$-rank at most $φ(d)/2-1$. For each $d\\in\\{4,6\\}$, both blocks vanish for a relative density-one set of primes $p\\equiv d+1\\pmod{2d}$; in particular, the even components vanish in accordance with Vandiver's conjecture. For each $d\\in\\{8,10,12\\}$, the odd block has order at most $p$ for a relative density-one set of primes in the same progression, so every summand $A_{aN}$, $a\\in U_d$, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet $L$-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves $A_{34}=0$ for every odd prime $p$; a single-file PARI/GP program reproduces the calculation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13932",
  "author_total": 2,
  "authors": [
    {
      "name": "Xuejun Guo"
    },
    {
      "name": "Zhengyu Tao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3pkrf7mxbk8y6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13932",
  "fulltext_url": null,
  "id": "3pkrf7mxbk8y6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13932",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7213317645",
  "pdf_url": "https://arxiv.org/pdf/2609.13932",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3pkrf7mxbk8y6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13932",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13932"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups",
  "title_text": "Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/3pkrf7mxbk8y6",
  "venue": "arXiv",
  "versions": []
}