{
  "abstract": "Let $k$ be an imaginary quadratic field and let $p \\ne 2$ be a prime number, split in $k$ into ${\\mathfrak p}{\\overline {\\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $δ\\geq 0$ be the ${\\mathfrak p}$-valuation of the ${\\overline {\\mathfrak p}}$-Fermat quotient of the fundamental ${\\mathfrak p}$-unit $x$ of $k$. Let $K/k$ be any bi-ramified ${\\mathbb{Z}}_p$-extension and let $p^e$ be the degree of the inertia field of ${\\mathfrak p}$, $\\overline {\\mathfrak p}$ being totally ramified. We prove that if $e \\geq δ$, then $λ(K/k) = 1$, $μ(K/k) = 0$; if $e<δ$ a characterization is obtained from Iwasawa invariants of the $S^{\\mathfrak p}$-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a $p$-adic regulator ${\\mathcal R}^{\\mathfrak p}_δ$ in incomplete $p$-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ${\\mathbb{Z}}_p$-extension is ''exceptional''; justifications are given. A pari/gp program computes $δ$ and ${\\mathcal R}^{\\mathfrak p}_δ$, for $p=3$, $e=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.09258",
  "author_total": 1,
  "authors": [
    {
      "name": "Georges Gras"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dcyqrps06j5hb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.09258",
  "fulltext_url": null,
  "id": "dcyqrps06j5hb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.09258",
  "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": "W7168136855",
  "pdf_url": "https://arxiv.org/pdf/2607.09258",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dcyqrps06j5hb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2607.09258",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.09258"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field",
  "title_text": "Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/dcyqrps06j5hb",
  "venue": "arXiv",
  "versions": []
}