Riemann hypothesis via absolute geometry - MathOverflow

Riemann hypothesis via absolute geometry

2011-07-03T10:00:54Z

Several leading mathematicians (e.g. Yuri Manin) have written or said publicly that there is a known outline of a likely natural proof of the Riemann hypothesis using absolute algebraic geometry over the field of one element; some like Mochizuki and Durov are thinking of a possible application of $\mathbf{F}_1$-geometry to an even stronger abc conjecture. It seems that this is one of the driving forces for studying algebraic geometry over $\mathbf{F}_1$ and that the main obstacle to materializing this proof is that the geometry over $\mathbf{F}_1$ (cf. MO what is the field with one element, applications of algebaric geometry over a field with one element) is still not satisfactorily developed. Even a longer-term attacker of the Riemann hypothesis from outside the algebraic geometry community, Alain Connes, has concentrated recently in his collaboration with Katia Consani on the development of a version of geometry over $\mathbf{F}_1$.

Could somebody outline for us the ideas in the folklore sketch of the proof of the Riemann hypothesis via absolute geometry ? Is the proof analogous to the Deligne's proof (article) of the Riemann-Weil conjecture (see wikipedia and MathOverflow question equivalent-statements-of-riemann-hypothesis-in-the-weil-conjectures) ?

Grothendieck was not happy with Deligne's proof, since he expected that the proof would/should be based on substantial progress on motives and the standard conjectures on algebraic cycles. Is there any envisioned progress in the motivic picture based on $\mathbf{F}_1$-geometry, or even envisioned extensions of the motivic picture ?

Answer by David Speyer for Riemann hypothesis via absolute geometry

2011-07-03T13:07:45Z

Warning: I am not an expert here but I'll give this a shot.

In the analogy between number fields and function field, Riemann's zeta funnction is the $\zeta$ function for $\mathrm{Spec} \ \mathbb{Z}$ . Note that $\mathrm{Spec} \ \mathbb{Z}$ is one dimensional. So proving the Riemann hypothesis should be like proving the Weil conjectures for a curve, which was done by Weil. Deligne's achievement was to prove the Weil conjectures for higher dimensional varieties which, according to this analogy, should be less relevant.

I wrote a blog post about one of the standard ways to prove the Riemann hypothesis for a curve $X$ (over $\mathbb{F}_p$). Note that a central role is played by the surface $X \times X$. I believe the $\mathbb{F}_1$ approach is to invent some object which can be called $(\mathrm{Spec} \ \mathbb{Z}) \times_{\mathbb{F}_1} (\mathrm{Spec} \ \mathbb{Z})$ .

Answer by S. Carnahan for Riemann hypothesis via absolute geometry

2011-07-04T04:42:40Z

Last fall, there was a conference in Nagoya about precisely this question (oddly enough, funded by a "Riemann Hypothesis" DARPA grant). Since I was attending a different conference at the same university at the same time, I didn't get to see all of the talks. However, Kedlaya's overview talk, which is listed among others on the schedule page, is rather informative.

Essentially, one hopes to get the completed $L$-function of an $\mathbb{F}_1$-scheme $X$ by cohomological means, by choosing a holomorphic family of operators (analogous to $1-q^{-s}\text{Frob}_q$ in the function field setting), and taking the determinant of the action on the cohomology of $X$ (which is expected to be infinite dimensional). This is basically a generalization of the Grothendieck-Lefschetz trace formula to a cohomology theory that is not yet known. There is some algebraic evidence that some form of the de Rham-Witt complex with a suitable alteration at infinity is such a cohomology theory, but I don't know what the appropriate family of operators ought to be. I am told that there are promising hints coming from the world of dynamical systems and foliated spaces, and this is where non-commutative geometry seems to enter the picture.