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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 ?

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    $\begingroup$ Just two comments. First, I am not sure that the abc-conjecture is in any way stronger than the Riemann Hypothesis. Second, I am skeptical about all algebraic or algebro-geometric attempts for the RH. People working in these fields often don't realize that automorphic L-functions conjecturally satisfy the RH (and have an Euler product, functional equation etc.), but most of them don't seem to be connected to any algebraic or algebro-geometric object, e.g. their coefficients look transcendental. In fact automorphicity seems to be the "reason" for RH, but we don't know how. $\endgroup$
    – GH from MO
    Commented Jul 3, 2011 at 16:03
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    $\begingroup$ @Zoran: I understand and I admire the successes of algebraic geometry, I just wanted to share some thoughts. BTW most automorphic L-functions (100% when you count in some density sense) have no clear connection to algebra or geometry. I don't know of any algebro-geometric framework that would be able or would only attempt to talk about the L-function of a full level Maass form, say. $\endgroup$
    – GH from MO
    Commented Jul 3, 2011 at 17:20
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    $\begingroup$ @GH: Are you who I think you are? Someone with initials GH made a similar comment to me a few years ago and it is an insightful comment. Anyway, hi! $\endgroup$
    – Felipe Voloch
    Commented Jul 3, 2011 at 18:35
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    $\begingroup$ @Felipe: I remember our conversation, it happened in a great Mexican restaurant! I have fond memories of those times. Of course it is possible that RH for Dirichlet L-functions or (say) for the L-functions of holomorphic newforms will be settled in the future by algebro-geometric means. That would be fantastic, of course, but it would (I think) leave the question open about the many transcendental looking automorphic L-functions out there. $\endgroup$
    – GH from MO
    Commented Jul 3, 2011 at 20:06
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    $\begingroup$ Alex, it is possible that at one level of development of mathematics things should be viewed as different phenomena, while on the other level, few centuries later, become reexplained by something now completely out of reach and intuition. $\endgroup$
    – Zoran Skoda
    Commented Jul 5, 2011 at 15:44

2 Answers 2

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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 $(\mathrm{Spec} \ \mathbb{Z}) \times_{\mathbb{F}_1} (\mathrm{Spec} \ \mathbb{Z})$.

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    $\begingroup$ David, that is already very helpful, but let me hear others' more detailed suggestions. One should also make sure that the usual Riemann hypothesis really reduces to that one (what will much depend on formalism) which you heuristically suggested. I know that otherwise nice algebraic $\mathbf{F}_1$-formalism of Durov suffers from the pathology that $Spec(\mathbf{Z})\times_{\mathbf{F}_1}Spec(\mathbf{Z})$ is not what one expects, but again $Spec(\mathbf{Z})$. $\endgroup$
    – Zoran Skoda
    Commented Jul 3, 2011 at 16:24
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    $\begingroup$ Quiaochu: there are several candidates for $\mathbf{F}_1$ and even $\mathbf{F}_{1^n}$ and for the corresponding wider framework accomodating them. Sphere spectrum is likely a base of again different (higher) geometry. But see some thoughts on a relation: mathoverflow.net/questions/1628/kf-1-sphere-spectrum $\endgroup$
    – Zoran Skoda
    Commented Jul 3, 2011 at 16:26
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    $\begingroup$ I definitely want to hear others suggestions as well. I don't know how one is supposed to complete the proof from here -- in particular, what object is supposed to play the role of the graph of Frobenius? $\endgroup$
    – David E Speyer
    Commented Jul 3, 2011 at 18:14
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    $\begingroup$ David is correct, as far as I know. There is no outline of a proof, just wishful thinking. In particular, the last comment about Frobenius is spot on. I just wanted to add that we may not need the full $Spec \mathbb{Z} \times_{\mathbb{F}_1} Spec \mathbb{Z}$ but maybe just a neighbourhood of the diagonal, as there are proofs of RH for function fields that only use a derivation and don't require higher dimensional objects. $\endgroup$
    – Felipe Voloch
    Commented Jul 3, 2011 at 18:29
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    $\begingroup$ Durov told me, that his approach yields that $Spec(Z)\times_{F_1}Spec(Z)=Spec Z$, which is certainly not very interesting. $\endgroup$
    – Mikhail Bondarko
    Commented Jul 3, 2011 at 21:23
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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.

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