On Bost-Connes type systems and complex multiplication
type: Item
instance of: scholarly article
title: On Bost-Connes type systems and complex multiplication
zbMATH Open document ID: 1253.11105
DOI: 10.4171/JNCG/92
author: Bora Yalkinoglu
published in: Journal of Noncommutative Geometry
publication date: 2012-06-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.0879
review text: In their fundamental paper [Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040)], \textit{J.-B. Bost} and \textit{A. Connes} constructed a quantum statistical mechanical system, known as the Bost-Connes or BC system, that recovers the explicit class field theory for \(\mathbb Q\). In this paper the author uses the theory of complex multiplication for Siegel modular varieties to construct arithmetic subalgebras for BC-type systems attached to number fields containing a CM field. The construction is based on the method used by \textit{E. Ha} and \textit{F. Paugam} [IMRP, Int. Math. Res. Pap. 2005, No. 5, 237--286 (2005; Zbl 1173.82305)] and extends the construction of arithmetic subalgebras for imaginary quadratic fields by \textit{A. Connes, M. Marcolli} and \textit{N. Ramachandran} [Sel. Math., New Ser. 11, No. 3-4, 325--347 (2005; Zbl 1106.58005)].
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11M55
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 81R60
Mathematics Subject Classification ID: 58B34
zbMATH DE Number: 6045439
zbMATH Keywords: Bost-Connes type systems
zbMATH Keywords: complex multiplication
zbMATH Keywords: Shimura varieties
MaRDI profile type: Publication
arXiv ID: 1010.0879
cites work: Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory
cites work: Q3854185
cites work: Q3913911
cites work: Q5440028
cites work: KMS states and complex multiplication
cites work: Q3424902
cites work: Q3877792
cites work: Q4821148
cites work: Q5473363
cites work: On Bost-Connes type systems for number fields
cites work: Q3409177
cites work: Automorphism Groups of Shimura Varieties and Reciprocity Laws
cites work: Q3998559
cites work: Q4002684
cites work: Q4510389
cites work: Q4317891
cites work: Q3884236
description: scientific article; zbMATH DE number 6045439