The “Arithmetic Site” of Connes and Consani

A week ago, Connes and Consani posted to the arXiv a 6 pages note titled The Arithmetic Site (and submitted to Elsevier owned CRAS).

The abstract ends with:

This note provides the algebraic geometric space underlying the non-commutative approach to RH.

which sounds important enough… And at the end of the introduction that same sentence is repeated, followed by:

It gives a geometric framework reasonably suitable to transpose the conceptual understanding of the Weil proof in finite characteristic as in [7]. This translation would require in particular an adequate version of the Riemann-Roch theorem in characteristic 1.

which sounds like there is still some distance before a proof of RH occurs (the reference [7] is this 1958 paper by Grothendieck, and wikipedia has some background for the notion of a site, and the Riemann-Roch theorem).

I’ve now noticed that this paper had been preceded by some lectures by Connes and Consani at Ohio State University.

Connes started working on a noncommutative geometry approach to RH in 1996 with a note in CRAS (freely available in Gallica) followed by this long 1998 paper. In recent years, with Consani and Marcolli, they have more and more evolved away from the physics interpretation side towards algebro-geometric notions such as the field with one element F_1, and tropical geometry.

If some readers with a good command of those topics wish to make informative comments on the latest note, they are very welcome to do so (comments are moderated but should appear within 24 hours).



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