The narrow class groups of the \(\mathbb Z_{17}\)- and \(\mathbb Z_{19}\)-extensions over the rational field
type: Item
instance of: scholarly article
title: The narrow class groups of the \(\mathbb Z_{17}\)- and \(\mathbb Z_{19}\)-extensions over the rational field
zbMATH Open document ID: 1214.11125
author: Kuniaki Horie
author: Mitsuko Horie
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2010-06-21T00:00:00Z
review text: Let \(p\) be a prime and \(\mathbb B_\infty\) the \(\mathbb Z_p\)-extension of \(\mathbb Q\). For brevity, let us denote the class-group of this field by \(C_p\), and the narrow class-group by \(C_p'\). Further, for any prime \(l\), \(C_p(l)\) and \(C_p'(l)\) will denote the \(l\)-parts of these groups. Iwasawa showed that \(C_p(p)\) vanishes; the article under review considers the vanishing of \(C_p(l)\) and \(C_p'(l)\) for \(l\neq p\) when \(p= 17\) or \(19\). In [[*] J. Lond. Math. Soc., II. Ser. 66, No. 2, 257--275 (2002; Zbl 1011.11072)], the first author showed that, given an arbitrary prime \(p\), if \(l\neq p\) is a prime that (a) is primitive mod \(p^2\) and (b) exceeds a certain explicit bound \(M_p\) depending on \(p\), then \(C_p(l)\) vanishes. The first author and both authors respectively built on this in [*] and [Acta Arith. 135, No. 2, 159--180 (2008; Zbl 1158.11046)] to remove the condition \(l> M_p\) when \(p\leq 13\). The main aim of the present article is to extend this to the primes \(17\) and \(19\). To summarize their results, they prove: Let \(p= 17\) or \(19\) and let \(l\) be a prime distinct from \(p\). If \(l\) is primitive mod \(p^2\) or \(l= 2\), then \(C_p'(l)\) vanishes. In particular, \(C_p(l)\) vanishes also. This is achieved by analysing the class number \(h_n\) of the \(n\)th field in the \(\mathbb Z_p\)-extension \(\mathbb B_\infty/\mathbb Q\), and in particular the integer \(h_n/h_{n-1}\). Indeed, they show that if \(l\) divides this integer, then \(l\) and \(n\) are bounded above by explicit integers. A part of their argument is to reduce to cases that can be readily checked on a computer, showing ultimately that \(l\) in fact cannot divide \(h_n/h_{n-1}\). Some remarks should be made: If \(l\) is odd, the vanishing of \(C_p'(l)\) is trivially equivalent to the vanishing of \(C_p(l)\), and the calculations are made on the latter. Otherwise, the authors instead use a result of \textit{J. V. Armitage} and \textit{A. Fröhlich} [Ma…
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5723789
zbMATH Keywords: \(\mathbb Z_p\)-extension
zbMATH Keywords: narrow class group
zbMATH Keywords: Iwasawa theory
reviewed by: Paul Buckingham
MaRDI profile type: Publication
full work available at URL: s12188-009-0030-3
OpenAlex ID: W2115722467
cites work: Classnumbers and unit signatures
cites work: Q5843509
cites work: Ideal Class Groups of Iwasawa-Theoretical Abelian Extensions Over the Rational Field
cites work: Primary components of the ideal class group of the \(\mathbb Z_p\)-extension over \(\mathbb Q\) for typical inert primes
cites work: Certain primary components of the ideal class group of the \(\mathbb Z_p\)-extension over the rationals
cites work: The narrow class groups of some Zp-extensions over the rationals
cites work: A note on class numbers of algebraic number fields
cites work: Class numbers and \(\mathbb Z_p\)-extensions
cites work: Q3943047
DOI: 10.1007/S12188-009-0030-3
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description: scientific article; zbMATH DE number 5723789