Greenberg's conjecture and the Iwasawa polynomial
type: Item
instance of: scholarly article
title: Greenberg's conjecture and the Iwasawa polynomial
zbMATH Open document ID: 0907.11038
published in: Journal of the Mathematical Society of Japan
publication date: 1999-02-25T00:00:00Z
review text: The conjecture attributed to Greenberg, claiming that all lambda invariants \(\lambda_p\) of real fields \(k\) are zero, is difficult to attack, and there are not too many results available. A lot of work has been done, but extensive numerical results seem to have been obtained mainly for the invariant \(\lambda_3\) and \(k\) a real quadratic field of discriminant \(D\), say. In this context, let us mention \textit{J. Kraft} and \textit{R. Schoof} [Compos. Math. 97, 135-155 (1995; Zbl 0840.11043)] and \textit{T. Fukuda} and \textit{H. Taya} [Acta Arith. 69, 277-292 (1995; Zbl 0828.11060)]. These articles show that \(\lambda_3\) is zero for all but a few discriminants \(D<10000\). The author of the article under review previously developed a simple criterion which however only applies in restricted circumstances. The essential condition one needed in order to use this criterion is that the characteristic polynomial of the Iwasawa module \(Y\) is irreducible. Here \(Y\) is the Galois group of the maximal \(p\)-ramified pro-\(p\)-extension \(M\) of \(k\). In the paper under review, the author extends this criterion by dint of clever and involved algebraic computations to a more general setting. For details, we have to refer to the paper, in particular Theorem 3. The new criterion enables the author to settle some more cases of \(D<10000\). Meanwhile, the author and \textit{H. Ichimura} have found a new approach which allows to do \textit{all} cases \(D<10000\) [Tohoku Math. J., II. Ser. 49, 203-215 (1997; Zbl 0886.11060) and Int. J. Math. 7, 721-744 (1996; Zbl 0881.11075)].
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 1185524
zbMATH Keywords: cyclotomic extensions
zbMATH Keywords: characteristic polynomials
zbMATH Keywords: Iwasawa polynomial
zbMATH Keywords: Greenberg conjecture
zbMATH Keywords: lambda invariants
Wikidata QID: Q122893690
author: Hiroki Sumida
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04940689
OpenAlex ID: W2094693905
DOI: 10.2969/JMSJ/04940689
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description: scientific article; zbMATH DE number 1185524