## Smirnov on $\mathbb{F}_1$ and the RH

Wednesday, Alexander Smirnov (Steklov Institute) gave the first talk in the $\mathbb{F}_1$ world seminar. Here’s his title and abstract:

Title: The 10th Discriminant and Tensor Powers of $\mathbb{Z}$

“We plan to discuss very shortly certain achievements and disappointments of the $\mathbb{F}_1$-approach. In addition, we will consider a possibility to apply noncommutative tensor powers of $\mathbb{Z}$ to the Riemann Hypothesis.”

Here’s his talk, and part of the comments section:

Smirnov urged us to pay attention to a 1933 result by Max Deuring in Imaginäre quadratische Zahlkörper mit der Klassenzahl 1:

“If there are infinitely many imaginary quadratic fields with class number one, then the RH follows.”

Of course, we now know that there are exactly nine such fields (whence there is no ‘tenth discriminant’ as in the title of the talk), and one can deduce anything from a false statement.

Deuring’s argument, of course, was different:

The zeta function $\zeta_{\mathbb{Q} \sqrt{-d}}(s)$ of a quadratic field $\mathbb{Q}\sqrt{-d}$, counts the number of ideals $\mathfrak{a}$ in the ring of integers of norm $n$, that is
$\sum_n \#(\mathfrak{a}:N(\mathfrak{a})=n) n^{-s}$
It is equal to $\zeta(s). L(s,\chi_d)$ where $\zeta(s)$ is the usual Riemann function and $L(s,\chi_d)$ the $L$-function of the character $\chi_d(n) = (\frac{-4d}{n})$.

Now, if the class number of $\mathbb{Q}\sqrt{-d}$ is one (that is, its ring of integers is a principal ideal domain) then Deuring was able to relate $\zeta_{\mathbb{Q} \sqrt{-d}}(s)$ to $\zeta(2s)$ with an error term, depending on $d$, and if we could run $d \rightarrow \infty$ the error term vanishes.

So, if there were infinitely many imaginary quadratic fields with class number one we would have the equality
$\zeta(s) . \underset{\rightarrow}{lim}~L(s,\chi_d) = \zeta(2s)$
Now, take a complex number $s \not=1$ with real part strictly greater that $\frac {1}{2}$, then $\zeta(2s) \not= 0$. But then, from the equality, it follows that $\zeta(s) \not= 0$, which is the RH.

To extend (a version of) the Deuring-argument to the $\mathbb{F}_1$-world, Smirnov wants to have many examples of commutative rings $A$ whose multiplicative monoid $A^{\times}$ is isomorphic to $\mathbb{Z}^{\times}$, the multiplicative monoid of the integers.

What properties must $A$ have?

Well, it can only have two units, it must be a unique factorisation domain, and have countably many irreducible elements. For example, $\mathbb{F}_3[x_1,\dots,x_n]$ will do!

(Note to self: contemplate the fact that all such rings share the same arithmetic site.)

Each such ring $A$ becomes a $\mathbb{Z}$-module by defining a new addition $+_{new}$ on it via
$a +_{new} b = \sigma^{-1}(\sigma(a) +_{\mathbb{Z}} \sigma(b))$
where $\sigma : A^{\times} \rightarrow \mathbb{Z}^{\times}$ is the isomorphism of multiplicative monoids, and on the right hand side we have the usual addition on $\mathbb{Z}$.

But then, any pair $(A,A’)$ of such rings will give us a module over the ring $\mathbb{Z} \boxtimes_{\mathbb{Z}^{\times}} \mathbb{Z}$.

It was not so clear to me what this ring is (if you know, please drop a comment), but I guess it must be a commutative ring having all these properties, and being a quotient of the ring $\mathbb{Z} \boxtimes_{\mathbb{F}_1} \mathbb{Z}$, the coordinate ring of the elusive arithmetic plane
$\mathbf{Spec}(\mathbb{Z}) \times_{\mathbf{Spec}(\mathbb{F}_1)} \mathbf{Spec}(\mathbb{Z})$

Smirnov’s hope is that someone can use a Deuring-type argument to prove:

“If $\mathbb{Z} \boxtimes_{\mathbb{Z}^{\times}} \mathbb{Z}$ is ‘sufficiently complicated’, then the RH follows.”

If you want to attend the seminar when it happens, please register for the seminar’s mailing list.

## The $\mathbb{F}_1$ World Seminar

For some time I knew it was in the making, now they are ready to launch it:

The $\mathbb{F}_1$ World Seminar, an online seminar dedicated to the “field with one element”, and its many connections to areas in mathematics such as arithmetic, geometry, representation theory and combinatorics. The organisers are Jaiung Jun, Oliver Lorscheid, Yuri Manin, Matt Szczesny, Koen Thas and Matt Young.

From the announcement:

“While the origins of the “$\mathbb{F}_1$-story” go back to attempts to transfer Weil’s proof of the Riemann Hypothesis from the function field case to that of number fields on one hand, and Tits’s Dream of realizing Weyl groups as the $\mathbb{F}_1$ points of algebraic groups on the other, the “$\mathbb{F}_1$” moniker has come to encompass a wide variety of phenomena and analogies spanning algebraic geometry, algebraic topology, arithmetic, combinatorics, representation theory, non-commutative geometry etc. It is therefore impossible to compile an exhaustive list of topics that might be discussed. The following is but a small sample of topics that may be covered:

Algebraic geometry in non-additive contexts – monoid schemes, lambda-schemes, blue schemes, semiring and hyperfield schemes, etc.
Arithmetic – connections with motives, non-archimedean and analytic geometry
Tropical geometry and geometric matroid theory
Algebraic topology – K-theory of monoid and other “non-additive” schemes/categories, higher Segal spaces
Representation theory – Hall algebras, degenerations of quantum groups, quivers
Combinatorics – finite field and incidence geometry, and various generalizations”

The seminar takes place on alternating Wednesdays from 15:00 PM – 16:00 PM European Standard Time (=GMT+1). There will be room for mathematical discussion after each lecture.

The first meeting takes place Wednesday, January 19th 2022. If you want to receive abstracts of the talks and their Zoom-links, you should sign up for the mailing list.

Perhaps I’ll start posting about $\mathbb{F}_1$ again, either here, or on the dormant $\mathbb{F}_1$ mathematics blog. (see this post for its history).

## Do we need the sphere spectrum?

Last time I mentioned the talk “From noncommutative geometry to the tropical geometry of the scaling site” by Alain Connes, culminating in the canonical isomorphism (last slide of the talk)

Or rather, what is actually proved in his paper with Caterina Consani BC-system, absolute cyclotomy and the quantized calculus (and which they conjectured previously to be the case in Segal’s Gamma rings and universal arithmetic), is a canonical isomorphism between the $\lambda$-rings
$\mathbb{Z}[\mathbb{Q}/\mathbb{Z}] \simeq \mathbb{W}_0(\overline{\mathbb{S}})$
The left hand side is the integral groupring of the additive quotient-group $\mathbb{Q}/\mathbb{Z}$, or if you prefer, $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ the integral groupring of the multiplicative group of all roots of unity $\mathbf{\mu}_{\infty}$.

The power maps on $\mathbf{\mu}_{\infty}$ equip $\mathbb{Z}[\mathbf{\mu}_{\infty}]$ with a $\lambda$-ring structure, that is, a family of commuting endomorphisms $\sigma_n$ with $\sigma_n(\zeta) = \zeta^n$ for all $\zeta \in \mathbf{\mu}_{\infty}$, and a family of linear maps $\rho_n$ induced by requiring for all $\zeta \in \mathbf{\mu}_{\infty}$ that
$\rho_n(\zeta) = \sum_{\mu^n=\zeta} \mu$
The maps $\sigma_n$ and $\rho_n$ are used to construct an integral version of the Bost-Connes algebra describing the Bost-Connes sytem, a quantum statistical dynamical system.

On the right hand side, $\mathbb{S}$ is the sphere spectrum (an object from stable homotopy theory) and $\overline{\mathbb{S}}$ its ‘algebraic closure’, that is, adding all abstract roots of unity.

The ring $\mathbb{W}_0(\overline{\mathbb{S}})$ is a generalisation to the world of spectra of the Almkvist-ring $\mathbb{W}_0(R)$ defined for any commutative ring $R$, constructed from pairs $(E,f)$ where $E$ is a projective $R$-module of finite rank and $f$ an $R$-endomorphism on it. Addition and multiplication are coming from direct sums and tensor products of such pairs, with zero element the pair $(0,0)$ and unit element the pair $(R,1_R)$. The ring $\mathbb{W}_0(R)$ is then the quotient-ring obtained by dividing out the ideal consisting of all zero-pairs $(E,0)$.

The ring $\mathbb{W}_0(R)$ becomes a $\lambda$-ring via the Frobenius endomorphisms $F_n$ sending a pair $(E,f)$ to the pair $(E,f^n)$, and we also have a collection of linear maps on $\mathbb{W}_0(R)$, the ‘Verschiebung’-maps which send a pair $(E,f)$ to the pair $(E^{\oplus n},F)$ with
$F = \begin{bmatrix} 0 & 0 & 0 & \cdots & f \\ 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$
Connes and Consani define a notion of modules and their endomorphisms for $\mathbb{S}$ and $\overline{\mathbb{S}}$, allowing them to define in a similar way the rings $\mathbb{W}_0(\mathbb{S})$ and $\mathbb{W}_0(\overline{\mathbb{S}})$, with corresponding maps $F_n$ and $V_n$. They then establish an isomorphism with $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ such that the maps $(F_n,V_n)$ correspond to $(\sigma_n,\rho_n)$.

But, do we really have the go to spectra to achieve this?

All this reminds me of an old idea of Yuri Manin mentioned in the introduction of his paper Cyclotomy and analytic geometry over $\mathbb{F}_1$, and later elaborated in section two of his paper with Matilde Marcolli Homotopy types and geometries below $\mathbf{Spec}(\mathbb{Z})$.

Take a manifold $M$ with a diffeomorphism $f$ and consider the corresponding discrete dynamical system by iterating the diffeomorphism. In such situations it is important to investigate the periodic orbits, or the fix-points $Fix(M,f^n)$ for all $n$. If we are in a situation that the number of fixed points is finite we can package these numbers in the Artin-Mazur zeta function
$\zeta_{AM}(M,f) = exp(\sum_{n=1}^{\infty} \frac{\# Fix(M,f^n)}{n}t^n)$
and investigate the properties of this function.

To connect this type of problem to Almkvist-like rings, Manin considers the Morse-Smale dynamical systems, a structural stable diffeomorphism $f$, having a finite number of non-wandering points on a compact manifold $M$.

From Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

In such a situation $f_{\ast}$ acts on homology $H_k(M,\mathbb{Z})$, which are free $\mathbb{Z}$-modules of finite rank, as a matrix $M_f$ having only roots of unity as its eigenvalues.

Manin argues that this action is similar to the action of the Frobenius on etale cohomology groups, in which case the eigenvalues are Weil numbers. That is, one might view roots of unity as Weil numbers in characteristic one.

Clearly, all relevant data $(H_k(M,\mathbb{Z}),f_{\ast})$ belongs to the $\lambda$-subring of $\mathbb{W}_0(\mathbb{Z})$ generated by all pairs $(E,f)$ such that $M_f$ is diagonalisable and all its eigenvalues are either $0$ or roots of unity.

If we denote for any ring $R$ by $\mathbb{W}_1(R)$ this $\lambda$-subring of $\mathbb{W}_0(R)$, probably one would obtain canonical isomorphisms

– between $\mathbb{W}_1(\mathbb{Z})$ and the invariant part of the integral groupring $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ for the action of the group $Aut(\mathbb{Q}/\mathbb{Z}) = \widehat{\mathbb{Z}}^*$, and

– between $\mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$ and $\mathbb{W}_1(\mathbb{Z}(\mathbf{\mu}_{\infty}))$ where $\mathbb{Z}(\mathbf{\mu}_{\infty})$ is the ring obtained by adjoining to $\mathbb{Z}$ all roots of unity.

## Alain Connes on his RH-project

In recent months, my primary focus was on teaching and family matters, so I make advantage of this Christmas break to catch up with some of the things I’ve missed.

Peter Woit’s blog alerted me to the existence of the (virtual) Lake Como-conference, end of september: Unifying themes in Geometry.

In Corona times, virtual conferences seem to sprout up out of nowhere, everywhere (zero costs), giving us an inflation of YouTubeD talks. I’m always grateful to the organisers of such events to provide the slides of the talks separately, as the generic YouTubeD-talk consists merely in reading off the slides.

Allow me to point you to one of the rare exceptions to this rule.

When I downloaded the slides of Alain Connes’ talk at the conference From noncommutative geometry to the tropical geometry of the scaling site I just saw a collage of graphics from his endless stream of papers with Katia Consani, and slides I’d seen before watching several of his YouTubeD-talks in recent years.

Boy, am I glad I gave Alain 5 minutes to convince me this talk was different.

For the better part of his talk, Alain didn’t just read off the slides, but rather tried to explain the thought processes that led him and Katia to move on from the results on this slide to those on the next one.

If you’re pressed for time, perhaps you might join in at 49.34 into the talk, when he acknowledges the previous (tropical) approach ran out of steam as they were unable to define any $H^1$ properly, and how this led them to ‘absolute’ algebraic geometry, meaning over the sphere spectrum $\mathbb{S}$.

Sadly, for some reason Alain didn’t manage to get his final two slides on screen. So, in this case, the slides actually add value to the talk…

## Closing in on Gabriel’s topos?

‘Gabriel’s topos’ (see here) is the conjectural, but still elusive topos from which the validity of the Riemann hypothesis would follow.

It is the latest attempt in Alain Connes’ 20 year long quest to tackle the RH (before, he tried the tools of noncommutative geometry and later those offered by the field with one element).

For the last 5 years he hopes that topos theory might provide the missing ingredient. Together with Katia Consani he introduced and studied the geometry of the Arithmetic site, and later the geometry of the scaling site.

If you look at the points of these toposes you get horribly complicated ‘non-commutative’ spaces, such as the finite adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^{\ast}$ (in case of the arithmetic site) and the full adele classes $\mathbb{Q}^*_+ \backslash \mathbb{A}_{\mathbb{Q}} / \widehat{\mathbb{Z}}^{\ast}$ (for the scaling site).

In Vienna, Connes gave a nice introduction to the arithmetic site in two lectures. The first part of the talk below also gives an historic overview of his work on the RH

The second lecture can be watched here.

However, not everyone is as optimistic about the topos-approach as he seems to be. Here’s an insightful answer on MathOverflow by Will Sawin to the question “What is precisely still missing in Connes’ approach to RH?”.

Other interesting MathOverflow threads related to the RH-approach via the field with one element are Approaches to Riemann hypothesis using methods outside number theory and Riemann hypothesis via absolute geometry.

About a month ago, from May 10th till 14th Alain Connes gave a series of lectures at Ohio State University with title “The Riemann-Roch strategy, quantizing the Scaling Site”.

The accompanying paper has now been arXived: The Riemann-Roch strategy, Complex lift of the Scaling Site (joint with K. Consani).

Especially interesting is section 2 “The geometry behind the zeros of $\zeta$” in which they explain how looking at the zeros locus inevitably leads to the space of adele classes and why one has to study this space with the tools from noncommutative geometry.

Perhaps further developments will be disclosed in a few weeks time when Connes is one of the speakers at Toposes in Como.