The BC-system and \(L\)-functions
type: Item
instance of: scholarly article
title: The BC-system and \(L\)-functions
zbMATH Open document ID: 1276.14002
author: Alain Connes
published in: Japanese Journal of Mathematics. 3rd Series
publication date: 2013-01-30T00:00:00Z
review text: This article is based on the 8th Takagi Lectures delivered by the author at the Research Institute for Mathematical Sciences of Kyoto University on November 2010. The paper surveys on some relations between \(L\)-functions and the Bost-Connes (BC-) system in noncommutative arithmetic geometry ([\textit{J.-B. Bost} and \textit{A. Connes}, Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040); \textit{A. Connes}, Sel. Math., New Ser. 5, No.1, 29--106 (1999; Zbl 0945.11015)]) also reporting on recent results obtained in collaboration with the reviewer [\textit{A.~Connes} and \textit{C.~Consani}, Compos. Math. 146, No. 6, 1383--1415 (2010; Zbl 1201.14001); \textit{A.~Connes} and \textit{C.~Consani}, in: G. van Dijk (ed.) et al., Casimir force, Casimir operators and Riemann hypothesis. Mathematics for innovation in industry and science. Proceedings of the conference, Fukuoka, Japan, November 9--13, 2009. Berlin: de Gruyter. 147--198 (2010; Zbl 1234.14002); \textit{A.~Connes} and \textit{C.~Consani}, J. Number Theory 131, No. 2, 159--194 (2011; Zbl 1221.14002); \textit{A.~Connes} and \textit{C.~Consani}, ``On the arithmetic of the BC-system'', \url{arXiv:1103.4672}, to appear in J. Noncommut. Geom.; \textit{A.~Connes} and \textit{C.~Consani}, ``Characteristic one, entropy and the absolute point'', in: Noncommutative Geometry, Arithmetic, and Related Topics, the Twenty-First Meeting of the Japan-U.S. Mathematics Institute, Baltimore 2009, JHUP. 75--139 (2012); \url{arXiv:0911.3537}] on the quest for an ``absolute arithmetic'' and on new relations of the BC-system with \(p\)-adic analysis and the theory of Witt vectors. The link between class field theory and type III factors in operators algebra that was discovered by the author several years ago is established by a quantum statistical mechanical system (the BC-system) that connects the theory of factors and the KMS condition in quantum mechanics through the implementation of a noncommutative space of class field …
reviewed by: Caterina Consani
Mathematics Subject Classification ID: 14A15
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11M55
Mathematics Subject Classification ID: 58B34
zbMATH DE Number: 6132926
zbMATH Keywords: BC-system
zbMATH Keywords: Witt rings
zbMATH Keywords: adèle class space
MaRDI profile type: Publication
full work available at URL: s11537-011-1035-0
OpenAlex ID: W1981084223
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DOI: 10.1007/S11537-011-1035-0
description: scientific article; zbMATH DE number 6132926