A note on Greenberg's conjecture for real abelian number fields
type: Item
instance of: scholarly article
title: A note on Greenberg's conjecture for real abelian number fields
zbMATH Open document ID: 0855.11056
DOI: 10.1007/BF02567825
author: Manabu Ozaki
author: Hisao Taya
published in: Manuscripta Mathematica
publication date: 1996-11-27T00:00:00Z
full work available at URL: https://eudml.org/doc/156130
review text: For an odd prime \(p\), let \(k\) be a real abelian field with \([k: \mathbb{Q}]\) prime to \(p\). According to Greenberg's conjecture, the Iwasawa invariant \(\lambda_p\) of the cyclotomic \(\mathbb{Z}_p\)-extension \(k_\infty\) of \(k\) should vanish. The authors present a criterion for the vanishing of the ``\(\chi\)-part'' of \(\lambda_p\), where \(\chi\) is a character of \(k\), under the assumption that the Iwasawa power series for the \(p\)-adic \(L\)-function \(L_p (s, \chi)\) is irreducible. In the case of a real quadratic field \(k= \mathbb{Q} (\sqrt {m})\) this leads to quite smooth criteria for \(\lambda_p =0\); for example, if \(p\) is either ramified or inert in \(k\), then \(\lambda_p\) vanishes if and only if the exponent of \(p\) in the \(p\)-adic regulator of \(k_n\) (the \(n\)th layer of \(k_\infty/ k\)) exceeds, for some \(n\geq 0\), an explicitly given bound depending only upon \(n\) and \(p\). As an application the authors give examples of \(\mathbb{Q} (\sqrt {m})\) with \(\lambda_3= 0\) for \(1< m< 10^4\), \(m\) squarefree and \(m\not\equiv 1\pmod 3\). The case in which 3 splits in \(\mathbb{Q} (\sqrt {m})\) has been studied by the second author and several other people; see, e.g., his joint article with \textit{T. Fukuda} [in Acta Arith. 69, No. 3, 277-292 (1995; Zbl 0828.11060)].
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 871770
zbMATH Keywords: real abelian number fields
zbMATH Keywords: class groups
zbMATH Keywords: Greenberg's conjecture
zbMATH Keywords: Iwasawa invariant
Wikidata QID: Q122957986
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871770