What is precisely still missing in Connes' approach to RH?

Question

I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps

Very roughly speaking the idea is to describe a dictionary which translates the concepts and techniques used in the proof of the analogue of the Riemann Hypothesis for function fields. This translation uses various techniques from tropical geometry and topos theory. At first I was hopeful I might understand the key issues with this translation, since I have some experience with the theory of Grothendieck toposes (or topoi). Nevertheless I have been lost when it comes to understanding precisely what the remaining problems are. As already briefly discussed in this thread: Riemann hypothesis via absolute geometry in the proof of the Riemann hypothesis for a curve $X$ over $F_p$ the surface $X \times X$ plays an important role. According to a new preprint of Connes / Consani there is now a suitable analogy for the surface $X \times X$ which is given by a topos called the "scaling site", cf. https://arxiv.org/abs/1603.03191

I would like to know what are the issues that are left open to complete the analogy with the proof of RH in the case of a curve over $F_p$?

Answer 1

First, recall the step's of Weil's proof, other than defining the surface:

1. Develop an intersection theory of curves on surfaces.
2. Show that the intersection of two specially chosen curves is equal to a coefficient of the zeta function.
3. Prove the Hodge index theorem using the Riemann-Roch theorem for surfaces (or is there another proof?).
4. By playing around with the two curves from step 2 and a few other simple curves, deduce an inequality for each coefficient of the logarithmic derivative of the zeta function.
5. Conclude the Riemann hypothesis for the roots of the zeta function.

Any approach which closely follows Weil's strategy would have to find analogues of these 5 steps.

I think most people would say the crux of the issue is step 3 - it would be very interesting if some abstract/categorical/algebraic method produced any inequality involving the Riemann zeta function or related functions that was not previously known, even if it was short of the full Riemann hypothesis. Equally, many people fear that if we generalize too far from the world of curves and surfaces that we know and love, we might lose the ability to prove concrete inequalities like this. nfdc23 discussed this issue in the comments.

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In the survey article you linke, Connes says:

At this point, what is missing is an intersection theory and a Riemann-Roch theorem on the square of the arithmetic site.

I think this is still the case - the Riemann-Roch theorem in the paper you link seems to be for certain curves in the scaling site. (If it were for the surface, I wouldn't expect to see it on the arXiv until the Riemann hypothesis were deduced from it, or a very convincing argument was found that it was not sufficient to prove the Riemann hypothesis!)

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I'm not sure how helpful knowledge of toposes (alone) will be for understanding this issue. The reason is that the key parts of any definition of intersection theory of curves on surfaces do not involve very much the topos structure of the surface, but rather other structures.

I know of a few different perspectives on intersection theory:

1. Add ample classes, apply a moving lemma, and then concretely count the intersections of curves. Here the main issues are global geometric considerations - ample divisors and their relationships to projective embeddings and hyperplanes and so on. I don't think toposes are a very useful tool for understanding these things.

2. Apply local formulas for the intersection number, and then sum over the point of intersection. This is more promising as e.g. Serre's formula is cohomological in nature, but it's the wrong kind of cohomology - for modules, not for toposes. I think it might be hard to define the right cohomology groups over semirings.

3. Take the cup product of their classes in a suitable cohomology theory. This at first looks promising, because one of our possible choices for a cohomology theory, etale cohomology, is defined using a topos. Unfortunately, because etale cohomology is naturally l-adic, it is difficult to establish positivity from it. Of course one could look to generalizing Deligne's proof of the Riemann hypothesis, but that is a different matter and much more complicated. (Peter Sarnak has suggested that Deligne's proof provides a guide to how one should try to prove the Riemann hypothesis for number fields, but it is not the etale topos but rather the use of fmailies in the proof that he wants to mimic.)

4. View one curve as an honest curve and the other as the divisor of a line bundle and pull the line bundle back to the curve, then compute its degree. This is, I think, the easiest way to define the intersection number, but one needs some of the other characterizations to prove its properties (even symmetry). This looks to me like it is at least potentially meaningful in some topos and I don't know exactly to what extent it can be applied to the arithmetic site.

Remember that even among locally ringed spaces, the ones where you have a good intersection theory are very specific (If you want to intersect arbitrary closed subsets I think you need smooth proper varieties over a field). For locally semiringed topoi, the situation is presumably much worse.

No one is saying it is impossible to develop an applicable intersection theory, but I don't think anyone really knows how to begin -- if they did, I'm sure they would be working furiously on it.

Answer 2

There is a very interesting book which appeared recently precisely on this topic: "The Riemann Hypothesis for Function Fields" by Machiel van Frankenhuijsen. His attempt at answering the OP's question about what is missing is in the last two chapters.

My impression is that Connes' work on RH, like that of great painters such as Picasso, can be divided into different periods. I would say the first period is about exploring connections to statistical mechanics and the notion of symmetry breaking. The main result is his article with Bost on the now called Bost-Connes system. However, this did not yet contain an approach towards RH. The second period is that of his Selecta paper with the trace formula. Finally the third period involves having "fun with $\mathbb{F}_1$", hyperreals, toposes, etc. The third period is about constructing suitable objects which would allow one to mimic Weil's proof. I am not familiar with this last development, but I think Will did a good job at explaining what is missing regarding the approach from this third period. The book I mentioned and the little I know concern the second period.

Let me first give a general somewhat vague opinion: if one has a reasonably clear description of what $A$ and $B$ are, then it is easier to prove a conjecture which says $A=B$ than one which says $A\neq B$ (or $A>B$, $A\ge B$, etc.). For example, I think the recent work by Kronheimer and Mrowka is significant progress on the four-color theorem. The latter says $A>B$ where $B=0$ and $A$ is the number of Tait colorings. They produced some invariant $C$ which they showed satisfies $C>B$ and reduced the problem to the conjecture $A=C$. Connes' Selecta article reduced Weil positivity (an $A\ge B$ type problem) and thus RH to an $A=B$ type problem, i.e., proving his global trace formula which is formally similar to a theorem of Guillemin and Sternberg for flows on Riemannian manifolds. He also proved the semi-local analogue of his formula. Take a finite group $G$ with a unitary representation $U$ and consider the matrix ${\rm tr}(U(gh^{-1}))$ with rows and columns indexed by $g,h\in G$. This matrix is positive semidefinite. Weil positivity follows from Connes' global trace formula for essentially the same reason.

Proving the trace formula, prima facie, looks like a problem of harmonic analysis or even constructive quantum field theory since one has a formally precise but still physicsy looking infinite-dimensional trace to make sense of by introducing UV and IR cutoffs and then analyzing the limit of removing them. There has been follow-ups by Haran (here and here) and Burnol (here and here) but, as far as I know, the trace formula has not been proved in this approach, even in the function field case where at least we know it should be true.

Answer 3

Terence Tao answered this question as follows:

" He (Connes) also discussed his recent paper with Consani in which they managed to view the (completed) Riemann zeta function as the Hasse-Weil zeta function of a curve over F_1, where F_1 is interpreted as the tropical ring {0,1}. So the Riemann hypothesis for the integers is now expressed in a very similar fashion to the Riemann hypothesis for curves, which has a number of proofs. Unfortunately, they are missing a huge ingredient in the dictionary, namely they have no Riemann-Roch theorem over F_1. Still, it is a very suggestive similarity, and something to keep an eye on in the future... "