# Durov approach to Arakelov geometry and $\mathbb{F}_1$

Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry over the tropical semiring. The analytic notions in Arakelov geometry seem to appear naturally, as a consequence of rephrasing the construction of $$\mathbb{Z}_p$$ in terms of maximal compact subgroups of $$GL(n, \mathbb{Q}_p)$$, and carrying this construction to $$GL(n, \mathbb{R})$$. A proposed construction of the field with one element $$\mathbb{F}_1$$ and its finite extensions also appear in the thesis.

Given these premises, I'd want to understand more of Durov's work. But Durov has left mathematics to found Telegram, and I am not sure about the status of his work. Since my background on arithmetic geometry is limited, and I am not currently in the university, I have troubles evaluating the impact of this approach, and I would like to know more before delving into a 568 pages thesis.

Has this theory been developed after he left mathematics? Did anyone find applications outside the theory itself? What is the point of view of people working in "classical" Arakelov geometry?

EDIT I had incorrectly assumed that after founding Telegram, Durov had left mathematics, but as @FedorPetrov points out, he is still active. Yet the question still stands: what is the status of this approach to Arakelov geometry? Is he (or other people) still developing it? Were there any results unrelated to generalized rings proved using this theory? I do not have access to his recent papers, but judging from the first page, his focus seems to have shifted somehow

• I do not think that he completely left mathematics. Not so often, but I meet him in the institute. He had a seminar joint with Alexander Leonidovich Smirnov, and they sometimes publish something on generalized rings in PDMI preprints. – Fedor Petrov Aug 12 '19 at 13:43
• @FedorPetrov that's interesting - do you have any links for the preprints? On the Arxiv I cannot find anything recent, and the search page on the PDMI site is in Russian, which I do not understand :-) – Andrea Ferretti Aug 12 '19 at 13:55
• Don't confuse Nikolai Durov (the author of the thesis) with his brother Pavel Durov (the founder of Telegram)! – Sasha Aug 12 '19 at 16:40
• I am not confusing them, they both founded Telegram together as far as I know (well, at least Wikipedia says so) – Andrea Ferretti Aug 12 '19 at 16:41
• @Sasha surely Nikolai is a key person in Telegram (and before that in Vkontakte). – Fedor Petrov Aug 12 '19 at 17:29

Let me just give a quick list of references and some brief comments:

There is a lot of work to do with generalized rings, especially concerning $$K$$-theoretic matters... All the usual approaches (via exact sequences and the like) seem to have failed to capture the information which is contained in these rings...

This being said, I am absolutely not working in this field, and would like to know a lot more myself! Thanks for reading!

• I should add that I just learned that Durov will be speaking at the Workshop on F1-Geometry and Motives for Alexander Smirnov's 60th birthday sites.google.com/view/smirnov60 – Andrea Ferretti Sep 12 '19 at 9:06

Regarding,

Did anyone find applications outside the theory itself?

The approaches of Durov (and a number of similar methods) are guided by getting an elegant philosophy, or getting some philosophies "right" already at the level of foundations. Also: Not all these ideas or aims are necessarily informed by Arakelov theory alone.

The state of the art is very far from being usable for attacking the kind of questions one could try to use classical Arakelov theory for (or, simpler, one could use Algebraic Number Theory for). One could, for example, demand that such an approach should give a novel (genuinely different) reproof for the finiteness theorems of classical number theory; but this so far has been lacking.

People have looked particularly at the $$K$$-theory because many arithmetic invariants can be extracted ("automatically") from $$K$$-theory, e.g. intersection theory (this is also the viewpoint frequently used in classical Arakelov theory, e.g. in Gillet-Soule; more classically unit group and class group of a number field also are found in $$K$$-theory). So if you get $$K$$-theory right, this should give you a whole supply of "right" definitions for free. To the best of my knowledge, although interesting things have been done, no miraculously useful information (for pragmatic number theorists) has yet been found in any of these viewpoints/computations. Of course K(F_1) being the sphere spectrum is aesthetically very pleasing, but it's still very far from giving you anything useful if you're interested in, say, a concrete Diophantine problem.

New ideas are needed. Maybe yours. Future will tell.