On \(p\)-adic zeta functions and \(\mathbb{Z}_p\)-extensions of certain totally real number fields
type: Item
instance of: scholarly article
title: On \(p\)-adic zeta functions and \(\mathbb{Z}_p\)-extensions of certain totally real number fields
zbMATH Open document ID: 0943.11049
author: Hisao Taya
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-09-04T00:00:00Z
review text: Let \(k\) be a totally real number field. For an odd prime \(p\) and for \(n \geq 0\), let \(A_n\) denote the Sylow \(p\)-group of the \(n\)th layer in the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\). Denote by \(A_n^\Gamma\) the Galois-invariant part of \(A_n\). The author proves that if \(p\) splits completely in \(k\) and Leopoldt's conjecture is true for \(k\) and \(p\), then card\((A_n ^\Gamma)\) equals the \(p\)-part of \(\zeta^*(1,k)\), for every sufficiently large \(n\), where \(\zeta^*(s,k)=\zeta (s,k)/ \zeta (s,\mathbb{Q})\), the quotient of the \(p\)-adic zeta functions of \(k\) and \(\mathbb{Q}\). Moreover, this \(p\)-part is equal to card\((A_0)\) times the \(p\)-part of \(R_p(k) - [k:\mathbb{Q}] + 1\), where \(R_p(k)\) denotes the \(p\)-adic regulator and \([k:\mathbb{Q}]\) the absolute degree of \(k\). This generalizes a result proved by \textit{T. Fukuda} and \textit{K. Komatsu} [J. Number Theory 23, 238-242 (1986; Zbl 0593.12003)] for real quadratic fields \(k\). As a consequence the author provides an alternative formulation for Greenberg's theorem on the nonvanishing of the Iwasawa invariants \(\lambda\) and \(\mu\) under the above assumptions. He also gives some computational data in the case \([k:\mathbb{Q}]=3\) and \(p=3\).
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 1321871
zbMATH Keywords: \(\mathbb{Z}_p\)-extensions
zbMATH Keywords: Iwasawa invariants
zbMATH Keywords: \(p\)-adic zeta functions
zbMATH Keywords: ideal class groups
zbMATH Keywords: totally real fields
zbMATH Keywords: Leopoldt's conjecture
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224850
OpenAlex ID: W2002045127
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DOI: 10.2748/TMJ/1178224850
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description: scientific article; zbMATH DE number 1321871