On the Iwasawa lambda invariant of an imaginary abelian field of conductor \(3p^{n+1}\)
type: Item
instance of: scholarly article
title: On the Iwasawa lambda invariant of an imaginary abelian field of conductor \(3p^{n+1}\)
zbMATH Open document ID: 1286.11175
author: Humio Ichimura
author: Shoichi Nakajima
author: Hiroki Sumida
published in: Journal of Number Theory
publication date: 2013-01-10T00:00:00Z
review text: In the paper under review the authors computationally verify certain cases of Greenberg's conjecture on the vanishing of Iwasawa's \(\lambda\)-invariant. For a number field \(K\) let \(\lambda_K\) be the `standard' Iwasawa \(\lambda\)-invariant of the cyclotomic \(\mathbb Z_3\)-extension of \(K\). Let \(p \geq 5\) be a prime and consider the fields \[ K_n = \mathbb Q(\mathrm{cos}(2 \pi / p^{n+1}), \zeta_3) \] and the maximal totally real subfields \(K_n^+\). Then it is shown that for \(5 \leq p \leq 599\) one has \[ \lambda_{K_n} = \lambda_{K_0}, \quad n \geq 0 \] and \[ \lambda_{K_n^+} = 0, \quad n \geq 0. \] The latter is as predicted by Greenberg's conjecture. A similar result in the case \(p=2\) is also established.
reviewed by: Andreas Nickel
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 6124195
zbMATH Keywords: Iwasawa invariant
zbMATH Keywords: Greenberg's conjecture
zbMATH Keywords: imaginary abelian field
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.08.022
OpenAlex ID: W2077176144
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DOI: 10.1016/J.JNT.2012.08.022
description: scientific article; zbMATH DE number 6124195