The Riemann–Roch strategy

Complex lift of the Scaling Site
  • Alain Connes
  • Caterina ConsaniEmail author


We describe the Riemann–Roch strategy which consists of adapting in characteristic zero Weil’s proof, of RH in positive characteristic, following the ideas of Mattuck–Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Riemann–Roch result over \({\mathbb C}\). In order to deal with arbitrary distribution functions this technique involves the results of Bohr, Jessen, and Tornehave on almost periodic functions.

Our main result is the construction, at the adelic level, of a complex lift of the adèle class space of the rationals. We interpret this lift as a moduli space of elliptic curves endowed with a triangular structure. The equivalence relation yielding the noncommutative structure is generated by isogenies. We describe the tight relation of this complex lift with the GL(2)-system. We construct the lift of the Frobenius correspondences using the Witt construction in characteristic 1.



The second named author would like to thank Alain Connes for introducing her to the noncommutative geometric vision on the Riemann Hypothesis and for sharing with her many mathematical ideas and insights.

Caterina Consani is partially supported by the Simons Foundation collaboration Grant no. 353677. She would like to thank Collège de France for some financial support.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.IHESBures sur YvetteFrance
  2. 2.Collège de France, Ohio State UniversityColumbusUSA
  3. 3.Johns Hopkins UniversityBaltimoreUSA

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