Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field
Abstract
Let k be an imaginary quadratic field and let p ≠ 2 be a prime number, split in k into pp̅. We assume that the p-class group of k is trivial. Let δ≥ 0 be the p-valuation of the p̅-Fermat quotient of the fundamental p-unit x of k. Let K/k be any bi-ramified ℤp-extension and let pe be the degree of the inertia field of p, p̅ being totally ramified. We prove that if e ≥ δ, then λ(K/k) = 1, μ(K/k) = 0; if e<δ a characterization is obtained from Iwasawa invariants of the Sp-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a p-adic regulator ℛpδ in incomplete p-ramification. It provides effective and computable results that complement some aspects of Iwasawa theory. Conjecture states that only the cyclotomic ℤp-extension is ''exceptional''; justifications are given. A pari/gp program computes δ and ℛpδ, for p=3, e=1.
Related
- Maximal unramified 3-extensions of imaginary quadratic fields and \textrm SL\sb 2(\mathbb Z\sb 3)
- Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies
- Ternary Perfect Forms Over Imaginary Quadratic Field
- p-class groups in the cyclotomic ℤp-extension of imaginary quadratic fields
- Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields
BibTeX
@misc{gras2026chevalleyherbrand,
title = {{Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field}},
author = {Georges Gras},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.09258},
eprint = {2607.09258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.09258},
}
RIS
TY - UNPB
TI - Chevalley-Herbrand formulas and Zp -extensions of a p-principal imaginary quadratic field
AU - Georges Gras
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2607.09258
UR - https://arxiv.org/abs/2607.09258
AB - Let k be an imaginary quadratic field and let p ≠ 2 be a prime number, split in k into pp̅. We assume that the p-class group of k is trivial. Let δ≥ 0 be the p-valuation of the p̅-Fermat quotient of the fundamental p-unit x of k. Let K/k be any bi-ramified ℤ_p-extension and let p^e be the degree of the inertia field of p, p̅ being totally ramified. We prove that if e ≥ δ, then λ(K/k) = 1, μ(K/k) = 0; if e<δ a characterization is obtained from Iwasawa invariants of the S^p-class groups. This approach only uses generalizations of Chevalley-Herbrand formulas and the non-nullity of a p-adic regulator ℛ^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 ℤ_p-extension is ''exceptional''; justifications are given. A pari/gp program computes δ and ℛ^p_δ, for p=3, e=1.
ER -