Greenberg's conjecture and relative unit groups for real quadratic fields
type: Item
instance of: scholarly article
title: Greenberg's conjecture and relative unit groups for real quadratic fields
zbMATH Open document ID: 0893.11044
author: Takashi Fukuda
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: Let \(p\) be an odd prime and let \(k\) be a real quadratic field. Let \(\mathbb{Q}\subset \mathbb{Q}_1\subset \dots\subset \mathbb{Q}_{\infty}\) and \(k\subset k_1\subset \dots\subset k_{\infty}\) be the \(\mathbb{Z}_p\)-extensions of \(\mathbb{Q}\) and \(k\). Let \(\sigma\) be a topological generator of Gal\((k_{\infty}/k)\). Let \(E(k_n)\) be the unit group of \(k_n\). The author defines the relative unit group of \(k_n\) to be \[ E_{n,R}=\{\varepsilon\in E(k_n) \mid N_{k_n/\mathbb{Q}_n}(\varepsilon)=\pm 1, N_{k_n/k}(\varepsilon)=\pm 1\} \] and defines \(E_{n,R,p^n}=\{\varepsilon\in E_{n,R} \mid \varepsilon^{1+\sigma}\in E_{n,R}^{p^n}\}\). Let \(V_n=E_{n,R,p^n}/E_{n,R}^{p^n}\). The author shows that \(V_n\) has order \(p^n\), and that \(V_n\) is cyclic if and only if there exists \(\varepsilon\in E_{n,R}\) such that \(\varepsilon\) and its conjugates generate a subgroup of \(E_{n,R}\) of index prime to \(p\). Assume now that \(p\) does not split in \(k\). Let \(A_n\) be the \(p\)-Sylow subgroup of the ideal class group of \(k_n\). Let \(i_{0,n}:A_0\to A_n\) be the natural map. A result of \textit{R. Greenberg} [Am. J. Math. 98, 263-284 (1996; Zbl 0334.12013)] says that the Iwasawa invariant \(\lambda_p(k)=0\) if and only if \(i_{0,n}\) is the zero map for some \(n\geq 1\). The present author shows that if \(i_{0,n}\) is injective then \(V_n\) is cyclic. Conversely, if \(V_n\) is cyclic for all \(n\geq 1\) then \(i_{0,n}\) is injective for all \(n\geq 1\). This result is used to give some examples where \(\lambda_3(k)=0\). Now assume that \(p\) splits in \(k\). Let \(D_n\) be the subgroup of \(A_n\) generated by powers of classes of the primes above \(p\). Then \(A_n=D_n\) for all \(n\geq 0\) if and only if \(V_n\) is cyclic for all \(n\geq 1\). If Greenberg's conjecture that \(\lambda_3(k)=0\) is true and \(V_n\) is cyclic for all \(n\geq 1\), then \(A_n=D_n\) for all \(n\geq 0\).
reviewed by: Lawrence C. Washington
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1039113
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Greenberg's conjecture
zbMATH Keywords: relative units
MaRDI profile type: Publication
OpenAlex ID: W2088239121
Wikidata QID: Q123011059
cites work: Bases normales, unités et conjecture faible de Leopoldt. (Normal bases, units, and the weak Leopoldt conjecture)
cites work: Cyclotomic units and Greenberg’s conjecture for real quadratic fields
cites work: On \({\mathbb{Z}}_ p\)-extensions of real quadratic fields
cites work: Normal bases and \(\lambda\)-invariants of number fields
cites work: A capitulation problem and Greenberg’s conjecture on real quadratic fields
cites work: Q4865351
cites work: The Iwasawa λ-invariants of ℤₚ-extensions of real quadratic fields
cites work: On the Iwasawa Invariants of Totally Real Number Fields
cites work: On the theory of cyclotomic fields
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: On Vandiver's conjecture and \({\mathbb{Z}}_ p\)-extensions of \({\mathbb{Q}}(\zeta_{p^ n})\)
cites work: Q4855224
cites work: Q5832112
cites work: A note on Greenberg's conjecture for real abelian number fields
cites work: Greenberg's conjecture and the Iwasawa polynomial
cites work: Computation of ℤ₃-invariants of real quadratic fields
DOI: 10.1006/JNTH.1997.2126
description: scientific article; zbMATH DE number 1039113