Twisted \(S\)-units, \(p\)-adic class number formulas, and the Lichtenbaum conjectures
type: Item
instance of: scholarly article
title: Twisted \(S\)-units, \(p\)-adic class number formulas, and the Lichtenbaum conjectures
zbMATH Open document ID: 0863.19003
DOI: 10.1215/S0012-7094-96-08421-5
author: Vincent Fleckinger
author: Manfred Kolster
author: Thong Nguyen Quang Do
published in: Duke Mathematical Journal
publication date: 1997-06-03T00:00:00Z
review text: This is an important paper in which delicate methods from Galois cohomology and Iwasawa theory are used, in conjunction with deep results in algebraic K-theory, to establish a version of the Lichtenbaum conjectures for absolutely abelian fields \(F\). These conjectures are far-reaching generalizations of the analytic class number formula, which will be recalled just below. Before we enter into the details of the paper under review, we'll try to give a general impression of what is going on. The Lichtenbaum conjectures are concerned with predictions about the order of even-numbered \(K\)-groups of the ring of integers \({\mathcal O}_F\) in a number field \(F\), and about regulators attached to odd-numbered \(K\)-groups. The philosophy is that even-numbered K-groups behave like ideal class groups, and the odd-numbered ones behave like unit groups. At the very least, one knows that \(K_{2i}({\mathcal O}_F)\) is finite, and \(K_{2i-1}({\mathcal O}_F)\) is finitely generated with known rank, by results of \textit{A. Borel}; \(K_0({\mathcal O}_F)\) is the direct sum of the ideal class group of \(F\) and a copy of \(\mathbb{Z}\), and \(K_1({\mathcal O}_F)\) is exactly the unit group of \({\mathcal O}_F\). Now the analytic class number formula provides information about the class group of \(F\), and the regulator of \(F\), which is a quantity attached to the unit group of \({\mathcal O}_F\). Actually, when one uses the functional equation of the Dedekind zeta function \(\zeta_F\), the more usual version of the analytic class number formula, involving the residue at \(s=1\), changes into the neater statement \(\zeta_F(0)^*=-h_FR_F/w_F\), where \(h_F\) and \(R_F\) are the class number and the regulator of \(F\), \(w_F\) counts the roots of unity in \(F\), and * means: take the first nonvanishing coefficient in the Taylor expansion at \(s=0\). This can be taken to be the Lichtenbaum conjecture for \(m=1\) (of course, it is a long-standing theorem). Let us now briefly recall th…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 943722
zbMATH Keywords: units
zbMATH Keywords: cyclotomic descent
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Galois cohomology
zbMATH Keywords: Lichtenbaum conjecture
zbMATH Keywords: absolutely abelian fields
zbMATH Keywords: generalization of analytic class number formula
zbMATH Keywords: Quillen \(K\)-group
zbMATH Keywords: étale cohomology
zbMATH Keywords: étale \(K\)-theory
zbMATH Keywords: Quillen-Lichtenbaum conjecture
Wikidata QID: Q123343973
MaRDI profile type: Publication
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