[Math] precisely still missing in Connes’ approach to RH

ag.algebraic-geometrynt.number-theory

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 $F_p$?

Best Answer

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.

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!)

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.

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