Generalised Iwasawa invariants and the growth of class numbers
type: Item
instance of: scholarly article
title: Generalised Iwasawa invariants and the growth of class numbers
zbMATH Open document ID: 1475.11198
author: Sören Kleine
published in: Forum Mathematicum
publication date: 2021-03-09T00:00:00Z
review text: Let \(K\) be a number field, let \(p\) be a rational prime and let \(d\) be a positive integer. Let \(\mathbb{K}\) be a \(\mathbb{Z}_{p}^{d}\)-extension of \(K\), that is, a Galois extension with \(\Gamma := \mathrm{Gal}(\mathbb{K}/K)\) topologically isomorphic to a direct product of \(d\) copies of the \(p\)-adic integers. For each positive integer \(n\), let \(\mathbb{K}_{n}\) be the subfield of \(\mathbb{K}\) fixed by \(\Gamma^{p^{n}}\). Let \(A_{n}\) denote the Sylow \(p\)-subgroup of the class group of the ring of integers of \(\mathbb{K}_{n}\) and define \(e_{n}\) by \(|A_{n}|=p^{e_{n}}\). In the case \(d=1\), \textit{K. Iwasawa} [Bull. Am. Math. Soc. 65, 183--226 (1959; Zbl 0089.02402)] showed that the growth of the order of \(A_{n}\) can be described in a very explicit manner: there exist integers \(n_{0},\lambda,\mu \geq 0\) and \(\nu\) such that for every \(n \geq n_{0}\), we have \(e_{n}=\mu p^{n} + \lambda n + \nu\). \textit{A. A. Cuoco} and \textit{P. Monsky} [Math. Ann. 255, 235--258 (1981; Zbl 0437.12003)] generalised this result to include the case \(d \geq 2\). They showed that there exist integers \(m_{0}, l_{0} \geq 0\), called the generalised Iwasawa invariants of \(\mathbb{K}/K\), such that \(e_{n}=(m_{0}p^{n} + l_{0}n + O(1))p^{(d-1)n}\). In the article under review, the author considers the local behaviour of generalised Iwasawa invariants on the set \(\mathcal{E}^{d}(K)\) of \(\mathbb{Z}_{p}^{d}\)-extensions of \(K\), with respect to a suitable topology. The main result is as follows. Let \(\mathbb{K}/K\) be a \(\mathbb{Z}_{p}^{d}\)-extension. Assume that there exists a prime of \(K\) that is totally ramified in \(\mathbb{K}/K\). Then with respect to a suitable topology on \(\mathcal{E}^{d}(K)\), there exists a neighbourhood \(\mathcal{U} \subseteq \mathcal{E}^{d}(K)\) of \(\mathbb{K}\) such that: \begin{itemize} \item[(i)] \(m_{0}(\mathbb{L}/K) \leq m_{0}(\mathbb{K}/K)\) for every \(\mathbb{L} \in \mathcal{U}\), and \item[(ii)] there exists …
reviewed by: Henri Johnston
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 7318794
zbMATH Keywords: generalised Iwasawa invariants
zbMATH Keywords: pseudo-null Iwasawa modules
zbMATH Keywords: growth of class numbers
zbMATH Keywords: boundedness of \(\lambda \)-invariants
zbMATH Keywords: Greenberg's generalised conjecture
MaRDI profile type: Publication
full work available at URL: forum-2019-0119
OpenAlex ID: W3084283667
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DOI: 10.1515/FORUM-2019-0119
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description: scientific article; zbMATH DE number 7318794