Connes, Alain and Consani, Caterina and Marcolli, Matilde (2009) Fun with F_1. Journal of Number Theory, 129 (6). pp. 1532-1561. ISSN 0022-314X http://resolver.caltech.edu/CaltechAUTHORS:20090714-095640002
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Abstract
We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field,” while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Video. For a video summary of this paper, please visit http:// www.youtube.com/watch?v=az_0pxm1jrI.
| Item Type: | Article | ||||||||
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| Additional Information: | © 2008 Elsevier Inc. Received 14 June 2008. Revised 1 August 2008. Available online 6 November 2008. Communicated by David Goss. The authors are partially supported by NSF grants DMS-FRG-0652164, DMS-0652431, and DMS- 0651925. The second author gratefully thanks l’Institut des Hautes Études Scientifiques for the hospitality, the pleasant atmosphere and the support received during a visit in January–April 2008. The third author would like to thank Abhijnan Rej for some useful conversations. | ||||||||
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| Subject Keywords: | BC-system; Endomotives; Varieties over F_1; Frobenius correspondence | ||||||||
| Record Number: | CaltechAUTHORS:20090714-095640002 | ||||||||
| Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20090714-095640002 | ||||||||
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| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||
| ID Code: | 14584 | ||||||||
| Collection: | CaltechAUTHORS | ||||||||
| Deposited By: | Tony Diaz | ||||||||
| Deposited On: | 22 Jul 2009 16:47 | ||||||||
| Last Modified: | 26 Dec 2012 11:05 |
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