The analogue of the Gauss class number problem in motivic cohomology
type: Item
instance of: scholarly article
title: The analogue of the Gauss class number problem in motivic cohomology
zbMATH Open document ID: 1350.11097
author: Caroline Junkins
author: Manfred Kolster
published in: Annales Mathématiques du Québec
publication date: 2014-09-30T00:00:00Z
full work available at URL: http://hdl.handle.net/11375/9119
review text: The classical Gauss problem is to determine all imaginary quadratic number fields of class number one. The paper under review studied the analogue of the Gauss class number problem in motivic cohomology. Let \(F\) be a number field and \(\zeta_F(s)\) be the Dedekind zeta function of \(F\). By Dirichlet's analytic class number formula and the functional equation, one has \[ \zeta_F(0)=-\frac{h(F)}{w(F)}, \] where \(h(F)\) is the class number of \(F\) and \(w(F)\) is the number of roots of unity in \(F\). In the paper under review, the authors study the analogous situation at negative integers. Let \(F\) be a totally real field with degree \(d\) and ring of integers \(O_F\). If we assume that the 2-primary Main Conjecture in Iwasawa theory is true for \(F\), the authors proved that, for \(n\geq 2\) even, \[ \zeta_F(1-n)=(-1)^{\frac{n}{2}}\frac{h_n(F)}{w_n(F)}, \] where \(h_n(F)\) is the order of the second motivic cohomology group \(H^2_\mathcal M(O_F,\mathbb Z(n))\), \(w_n(F)\) is the order of the Galois cohomology group \(H^0(F,\mathbb Q/\mathbb Z(n))\). Note that we have a surjective homomorphism [\textit{M. Kolster}, Math. Ann. 323, No. 4, 667--692 (2002; Zbl 1007.11068)] \[ H^2_\mathcal M(O_F,\mathbb Z(n))\rightarrow (\mathbb Z/2\mathbb Z)^d. \] The analogue of the classical Gauss class number problem is to determine all totally real number fields \(F\) with \(2^{-d}h_n(F)=1\). The authors showed that there are no such fields for \(n\geq 6\), there is only the field \(\mathbb Q(\sqrt{5})\) for \(n=4\), and there are 11 fields for \(n=2\). The \textit{motivic Gauss problem} is to determine the totally real fields \(F\) and even integers \(n\geq 2\) with \(h_n^\mathcal M(F)=1\), where \(h_n^\mathcal M(F)\) is the order of the so called \textit{motivic wild kernel} \(WK_{2n-2}^\mathcal M(F)\), which is the kernel of the homomorphism \[ H^2_\mathcal M(F,\mathbb Z(n))\rightarrow\bigoplus_{v\text{ non-complex}}H^2_\mathcal M(F_v,\mathbb Z(n)). \] The authors show that …
reviewed by: Hourong Qin
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11Y40
Mathematics Subject Classification ID: 14F42
zbMATH DE Number: 6350055
zbMATH Keywords: special values of zeta-functions
zbMATH Keywords: Gauss problem
zbMATH Keywords: motivic cohomology
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: Iwasawa theory
describes a project that uses: PARI/GP
describes a project that uses: SageMath
MaRDI profile type: Publication
OpenAlex ID: W2076670772
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DOI: 10.1007/S40316-013-0006-7
description: scientific article; zbMATH DE number 6350055