Class numbers in \(\mathbb Z^ d_ p\)-extensions. III
type: Item
instance of: scholarly article
title: Class numbers in \(\mathbb Z^ d_ p\)-extensions. III
zbMATH Open document ID: 0596.12005
DOI: 10.1007/BF01160471
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173801
review text: Let \(L/k\) be a multiple \(\mathbb Z_ p\)-extension of a number field and let \(e_ n=e_ n(L/k)\) be the exponent to which \(p\) appears in the class number of the \(n\)th layer, \(k_ n\), of the extension. In a previous paper [ibid. 191, 377--395 (1986; Zbl 0563.12004)] the author defined non-negative integers \(a=a(L/k)\) and \(\lambda^*=\lambda^*(L/k)\) and showed that \(e_ n=(\lambda^*n+O(1))p^{an}\). The present paper makes a deeper study of the growth of \(e_ n\) when \(a(L/k)=1\). The results are: (a) Let \(H_ n\) be the ideal class group of \(k_ n\). Then for each fixed \(j\geq 0\) the \(\mathbb Z/p\)-dimension of \(p^ jH_ n/p^{j+1}H_ n\) is \(c_ jp^{n-j}+O(1)\) for some integer \(c_ j\). (b) Suppose that \(\lambda^*(L/k)=0\). Then \(\sum^{\infty}_{0}c_ jp^{-j}\) converges to a positive rational \(\alpha\) and \(e_ n\sim \alpha p^ n\). More precisely, \(e_ n\leq \alpha p^ n+O(n^ 2)\) while \(e_ n\geq \alpha p^ n-O(p^{n/2}).\) In fact purely algebraic generalizations of (a) and (b) are established. To this end the author considers a complete local ring (\({\mathcal O},{\mathfrak m})\) whose reduction mod \(p\) has altitude 1, a finitely generated \(\mathbb Z_ p\)-submodule \(E\) of the multiplicative group \(1+{\mathfrak m}\), and a finitely generated \({\mathcal O}\)-module \(M\). He takes \(I_ n\) to be the ideal of \({\mathcal O}\) generated by the \(u^{p^ n}-1\), \(u\in E\). Then analogues of (a) and (b) hold where \(\mathbb Z/p\) is replaced by \({\mathcal O}/{\mathfrak m}\), \(H_ n\) by \(M/I_ nM\) (or some slightly more complicated \({\mathcal O}\)-module), and \(e_ n\) by the \({\mathcal O}\)-length of the \(p\)-power torsion subgroup of \(M/I_ nM\). The \(c_ j\) may be defined as sums of local contributions, \(c_ j({\mathfrak P})\), one for each minimal prime \({\mathfrak P}\) of p\({\mathcal O}\). The definition of \(c_ j({\mathfrak P})\) is made to rest on the following simple facts. First, the integral closure of \({\mathcal O}/{\mathfrak P}\) is …
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 3959566
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: multiple \({\mathbb{Z}}_ p\)-extension
zbMATH Keywords: class number
zbMATH Keywords: ideal class group
zbMATH Keywords: complete local ring
zbMATH Keywords: complete \({\mathbb{Z}}_ p\)-group algebra
zbMATH Keywords: Greenberg-Iwasawa module
MaRDI profile type: Publication
cites work: The Hilbert-Kunz function
cites work: \(p\)-ranks of class groups in \(Z^n_p\)-extensions
cites work: Class numbers in \({\mathbb{Z}}^ d_ p\)-expansions. II
OpenAlex ID: W2783754152
author: Paul Monsky
description: scientific article; zbMATH DE number 3959566