On the Iwasawa invariants of certain real abelian fields
type: Item
instance of: scholarly article
title: On the Iwasawa invariants of certain real abelian fields
zbMATH Open document ID: 0886.11060
author: Humio Ichimura
author: Hiroki Sumida
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-04-23T00:00:00Z
review text: Let \(k\) be a real abelian field and let \(p\) be an odd prime that does not split in \(k\). Greenberg's conjecture predicts that the Iwasawa invariant \(\lambda_p\) of \(k\) is zero. Assuming some further conditions for the pair \((k,p)\), the authors find a necessary and sufficient condition for this to be true. For many classes of abelian fields their criterion can be transformed into a form which involves only rational arithmetic. In particular, such a rational criterion is provided for certain real quadratic fields. For these fields there exist extensive previous calculations of \(\lambda_3\); the authors are able to include some new cases, and they also calculate values of \(\lambda_5\) and \(\lambda_7\). To get an idea of the criterion, consider the \(n\)th layer (\(n\geq 0\)), say \(k_n\), of the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\). Let \(E_n\) and \(\mathcal U_n\) denote the unit group of \(k_n\) and the group of principal units in the (unique) non-archimedean completion of \(k_n\), respectively. Denote by \(\mathcal E_n\) the closure of \(E_n \cap \mathcal U_n\) in \(\mathcal U_n\). As usual, regard \(\mathcal E_n\) and \(\mathcal U_n\) as modules over \(\mathbb{Z}_pT\) and consider their components \(\mathcal E_n(\chi)\) and \(\mathcal U_n(\chi)\) for the even characters \(\chi\) of \(k\). Let \(\lambda_\chi\) denote the \(\chi\)-component of \(\lambda_p\). The authors' result is based on the fact that \(\lambda_\chi\) vanishes if and only if, for some \(n\), \(\mathcal U_n(\chi)\) properly contains \(\mathcal E_n(\chi)\). To have control of \(\mathcal U_n(\chi)\) they give an explicit generator for it; this comes from a cyclotomic unit \(c_n\) defined by means of the norm of \(1 - \zeta _{f_n}\), where \(f_n\) is the conductor of \(k_n\) (\(\zeta _m\) is a primitive \(m\)th root of 1). The criterion is formulated in terms of \(c_n\), and its rational variant depends on an auxiliary prime \(l \equiv 1 \pmod {f_n}\). This variant resembles …
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 1054463
zbMATH Keywords: Iwasawa invariants
zbMATH Keywords: abelian fields
zbMATH Keywords: quadratic fields
zbMATH Keywords: Greenberg's conjecture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225147
OpenAlex ID: W1995764614
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DOI: 10.2748/TMJ/1178225147
description: scientific article; zbMATH DE number 1054463