\(\mathcal S\)-units and \(\mathcal S\)-class groups of a cyclic number field of prime degree
type: Item
instance of: scholarly article
title: \(\mathcal S\)-units and \(\mathcal S\)-class groups of a cyclic number field of prime degree
zbMATH Open document ID: 0958.11073
published in: Journal of Number Theory
publication date: 2001-01-01T00:00:00Z
review text: If \(L/K\) is a Galois extension of number fields with group \(G\), then the \(G\)-structures of the unit group \(E_L\) and the class group \(Cl_K\) are (conjecturally) linked. This is made more precise in Chinburg's Third Conjecture, and in the yet sharper conjecture dubbed Lifted Root Number Conjecture, or also Equivariant Tamagawa Number Conjecture. These conjectures are far-reaching and fascinate by their generality; it can be hard, however, to extract concrete isomorphism statements from them. The merit of the present article is that it presents such concrete results, with a moderate expenditure of technique. To do so, the author makes the (of course rather restrictive) assumption that \(K\) is the field of rationals and \(G\) is of prime order \(l\). This assumption has the advantage that torsion-free \(ZG\)-modules (``lattices'') are completely classified by the theorem of Diederichsen and Reiner. (The first of these two names is persistently misspelled in the paper.) These lattices are given by some discrete invariants (three natural numbers) and one element in the class group of \(\mathbb Q(\zeta_l)\), the so-called class invariant. Theorem 3.1 gives the discrete invariants not only for \(E_K\) but for all \(S\)-unit groups \(E_{K,S}\) as well. (Actually for \(E_K\) itself the result is not surprising.) The class invariant of \(E_{K,S}\) is subtler and cannot be calculated right off. Instead, the author deduces from the Gras ``conjecture'' (which is a theorem, proved from the Main Conjecture) that it equals the class invariant of the \(S\)-class group of \(K\). The tools used include cohomology and the theory of ambiguous classes. The article contains enlightening remarks concerning the question whether we are able to exhibit an example in the given context where \(U_K = E_K/\{\pm 1\}\) is not free over \(Z[\zeta_l]\). Here \(l\) would have to be at least 23, and in spite of an optimistic remark of Brumer, even with the aid of computers no certified counter…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1547372
zbMATH Keywords: Galois modules
zbMATH Keywords: Reiner's theorem
zbMATH Keywords: units
zbMATH Keywords: class groups
author: Isabelle Dubois
MaRDI profile type: Publication
full work available at URL: jnth.2000.2519
OpenAlex ID: W2067943529
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DOI: 10.1006/JNTH.2000.2519
description: scientific article; zbMATH DE number 1547372