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The Weil Proof and the Geometry of the Adelès Class Space

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Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 269))

Summary

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.

2000 Mathematics Subject Classifications: 11M36, 58B34, 11M26, 46L55

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Author information

Authors and Affiliations

  1. Collège de France, 3, rue d’Ulm, Paris, F-75005, France

    Alain Connes

  2. Mathematics Department, Johns Hopkins University, Baltimore, MD, 21218, USA

    Caterina Consani

  3. Department of Mathematics, California Institute of Technology 1200 E California Blvd, Pasadena, CA, 91101, USA

    Matilde Marcolli

Authors
  1. Alain Connes
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  2. Caterina Consani
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  3. Matilde Marcolli
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Correspondence to Alain Connes .

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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

  2. Eberly College of Science, Dept. Mathematics, Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Yuri Zarhin

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Connes, A., Consani, C., Marcolli, M. (2009). The Weil Proof and the Geometry of the Adelès Class Space. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_8

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