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The Weil Proof and the Geometry of the Adelès Class Space

Connes, Alain and Consani, Caterina and Marcolli, Matilde (2010) The Weil Proof and the Geometry of the Adelès Class Space. In: Algebra, Arithmetic, and Geometry. Volume I: In Honor of Yu. I. Manin. Progress in Mathematics. No.269. Springer , Boston, MA, pp. 339-405. ISBN 978-0-8176-4744-5. https://resolver.caltech.edu/CaltechAUTHORS:20170713-091114280

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Abstract

This paper explores analogies between the Weil proof of the Riemann hypothesis for function fields and the geometry of the adèles class space, which is the noncommutative space underlying Connes' spectral realization of the zeros of the Riemann zeta function. We consider the cyclic homology of the cokernel (in the abelian category of cyclic modules) of the “restriction map” defined by the inclusion of the idèles class group of a global field in the noncommutative adèles class space. Weil's explicit formula can then be formulated as a Lefschetz trace formula for the induced action of the idèles class group on this cohomology. In this formulation the Riemann hypothesis becomes equivalent to the positivity of the relevant trace pairing. This result suggests a possible dictionary between the steps in the Weil proof and corresponding notions involving the noncommutative geometry of the adèles class space, with good working notions of correspondences, degree, and codegree etc. In particular, we construct an analog for number fields of the algebraic points of the curve for function fields, realized here as classical points (low temperature KMS states) of quantum statistical mechanical systems naturally associated to the periodic orbits of the action of the idèles class group, that is, to the noncommutative spaces on which the geometric side of the trace formula is supported.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-0-8176-4745-2_8DOIArticle
https://arxiv.org/abs/math/0703392arXivDiscussion Paper
Additional Information:© 2009 Springer Science+Business Media, LLC. First Online: 21 November 2010. Dedicated to Yuri Manin on the occasion of his 70th birthday.
Subject Keywords:noncommutative geometry, cyclic homology, thermodynamics, factors, Riemann zeta function, correspondences, Lefschetz trace formula, adèles, idèles, Frobenius
Series Name:Progress in Mathematics
Issue or Number:269
Classification Code:2000 Mathematics Subject Classifications: 11M36, 58B34, 11M26, 46L55
Record Number:CaltechAUTHORS:20170713-091114280
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170713-091114280
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79068
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Jul 2017 16:19
Last Modified:03 Oct 2019 18:15

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