On the Iwasawa invariants of totally real number fields
type: Item
instance of: scholarly article
title: On the Iwasawa invariants of totally real number fields
zbMATH Open document ID: 0796.11048
DOI: 10.1007/BF02568324
author: Hiroshi Yamashita
published in: Manuscripta Mathematica
publication date: 1994-09-29T00:00:00Z
full work available at URL: https://eudml.org/doc/155820
review text: The author finds conditions guaranteeing the vanishing of the Iwasawa invariants \(\lambda\) and \(\mu\), for an odd prime \(\ell\), of a totally real number field \(k\). Let \(K= k(e^{2\pi i/\ell})\) and denote by \(k_ n\) and \(K_ n\) the \(n\)th layers of the cyclotomic \(\mathbb{Z}_ \ell\)- extensions of \(k\) and \(K\), respectively \((n\geq 1)\). For \(K_ n\), let \(C_ n\) denote the \(\ell\)-class group factored by the subgroup generated by all ideals whose prime factors divide \(\ell\). In the group ring \(\mathbb{Z}_ \ell [\text{Gal}(K/k)]\) let \(\varepsilon\) be the idempotent determined by the Teichmüller character. The main result is that \(\lambda= \mu=0\) if either of the following conditions (A) and (B) is satisfied; (A) \(\varepsilon C_ 1= \{1\}\) and no prime dividing \(\ell\) decomposes completely in \(K/k\); (B) there is \(m\geq 1\) such that \(\varepsilon C_{m+1}= \{1\}\) and no prime dividing \(\ell\) decomposes in \(K_{m+1}/ K_ m\). Under the condition (B) it is also proved that the Leopoldt conjecture holds for \(\ell\) in \(k_ n\) for every \(n\). The proofs are based on results by G. Gras and J.-F. Jaulent; see, in particular, \textit{J.-F. Jaulent's} thesis [Publ. Math. Fac. Sci. Besançon, Théor. Nombres Années 1984/1985-1985/1986, No. 1, 349 p. (1986; Zbl 0601.12002)].
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 492773
zbMATH Keywords: cyclotomic extension
zbMATH Keywords: Iwasawa invariants
zbMATH Keywords: Leopoldt conjecture
MaRDI profile type: Publication
cites work: On regular number fields
cites work: Onp-adicL-functions and cyclotomic fields. II
cites work: Représentations \(\ell\)-adiques associées aux invariants cyclotomiques. (\(\ell\)-adique representations associated with cyclotomic invariants.)
cites work: On the class groups of imaginary abelian fields
cites work: Q3943047
Recommended article: Q4541490
Recommended article: On p-adic L-functions and the Riemann-Hurwitz genus formula
Recommended article: On \(p\)-adic zeta functions and \(\mathbb{Z}_p\)-extensions of certain totally real number fields
Recommended article: Genus formulas and Greenberg's conjecture
Recommended article: A criterion for Greenberg's conjecture
Recommended article: On the Iwasawa \(\lambda\)-invariants of real quadratic fields
Recommended article: On the Iwasawa invariants of certain real abelian fields
Recommended article: Transformation de Mellin-Leopoldt des fonctions elliptiques. (Mellin- Leopoldt transform of elliptic functions)
Recommended article: A conditional proof of the Leopoldt conjecture for CM fields
Recommended article: Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields
description: scientific article; zbMATH DE number 492773