\(K_2\) and the Greenberg conjecture in multiple \(\mathbb Z_p\)-extensions
type: Item
instance of: scholarly article
title: \(K_2\) and the Greenberg conjecture in multiple \(\mathbb Z_p\)-extensions
zbMATH Open document ID: 1091.11041
author: David Vauclair
author: Thong Nguyen Quang Do
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/52469
review text: The ``generalized Greenberg conjecture'' (GG) affirms that the usual Iwasawa module \(X(\widetilde K)\) attached to the compositum \(\widetilde K\) of all \(\mathbb Z_p\)-extensions of a given number field \(K\) is pseudo-null over the Iwasawa algebra \(\widetilde \Lambda\) (which is defined as \(\mathbb Z_pG(\widetilde K/K)\)). Pseudo-nullity means that the module has support of codimension at least 2. The Galois group \(G(\widetilde K/K)\) is isomorphic to \(\mathbb Z_p^d\) for some \(d\). If \(K\) is totally real and Leopoldt's conjecture holds for \(K\) and \(p\), then \(d=1\); we get back Greenberg's usual conjecture. (A finitely generated \(\mathbb Z_pT\)-module is pseudo-null iff it is finite.) \textit{W. G. McCallum} [Am. Math. J. 123, No. 5, 909--939 (2001; Zbl 1020.11068)] proposed a criterion for (GG) to hold, which would include for instance the field \(K\) mentioned at the end of this review. As explained in the paper under review, his argument seems to contain a gap which has not be fixed so far. (Note that the reference number in line 7 of p .670 should be [15], not [13]). The authors therefore change the strategy somewhat and come up with a slightly different approach. They proceed in three steps, the first two of which are essentially known from the work of McCallum: (1) (GG) is equivalent to the triviality of the \(\widetilde \Lambda\)-torsion submodule \(T=\text{tors}_{\widetilde\Lambda}{\mathcal X}(\widetilde K)\), where \({\mathcal X}(M)\) denotes the Galois group of the maximal abelian pro-\(p\)-extension of \(M\) unramified outside \(p\). (2) One looks at the codescent map \(\delta\) from \(T_{\widetilde \Gamma}\) to \({\mathcal X}(K)\) and proves its injectivity. (3) One proves that the map \(\delta\) is null. This together with (2) proves the vanishing of \(T\) (even before taking coinvariants), and (GG) follows by virtue of (1). Actually one may replace \(\mathcal X\) and \(\delta\) by twisted versions \({\mathcal X}^{(i)}\) and \(\…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5016580
zbMATH Keywords: class groups
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: cohomology
zbMATH Keywords: capitulation
MaRDI profile type: Publication
OpenAlex ID: W136267758
Wikidata QID: Q122980672
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DOI: 10.5802/JTNB.513
description: scientific article; zbMATH DE number 5016580