Previously, we have recalled comparisons between approaches to define a geometry over the absolute point and art-historical movements, first those due to Yuri I. Manin, subsequently some extra ones due to Javier Lopez Pena and Oliver Lorscheid.

In these comparisons, the art trend appears to have been chosen more to illustrate a key feature of the approach or an appreciation of its importance, rather than giving a visual illustration of the varieties over $\mathbb{F}_1$ the approach proposes.

Some time ago, we’ve had a couple of posts trying to depict noncommutative varieties, first the illustrations used by Shahn Majid and Matilde Marcolli, and next my own mental picture of it.

In this post, we’ll try to do something similar for affine varieties over the absolute point. To simplify things drastically, I’ll divide the islands in the Lopez Pena-Lorscheid map of $\mathbb{F}_1$ land in two subsets : the former approaches (all but the $\Lambda$-schemes) and the current approach (the $\Lambda$-scheme approach due to James Borger).

The former approaches : Francis Bacon “The Pope” (1953)

The general consensus here was that in going from $\mathbb{Z}$ to $\mathbb{F}_1$ one looses the additive structure and retains only the multiplicative one. Hence, ‘commutative algebras’ over $\mathbb{F}_1$ are (commutative) monoids, and mimicking Grothendieck’s functor of points approach to algebraic geometry, a scheme over $\mathbb{F}_1$ would then correspond to a functor

$h_Z~:~\mathbf{monoids} \longrightarrow \mathbf{sets}$

Such functors are described largely by combinatorial data (see for example the recent blueprint-paper by Oliver Lorscheid), and, if the story would stop here, any Rothko painting could be used as illustration.

Most of the former approaches add something though (buzzwords include ‘Arakelov’, ‘completion at $\infty$’, ‘real place’ etc.) in order to connect the virtual geometric object over $\mathbb{F}_1$ with existing real, complex or integral schemes. For example, one can make the virtual object visible via an evaluation map $h_Z \rightarrow h_X$ which is a natural transformation, where $X$ is a complex variety with its usual functor of points $h_X$ and to connect both we associate to a monoid $M$ its complex monoid-algebra $\mathbb{C} M$. An integral scheme $Y$ can then be said to be ‘defined over $\mathbb{F}_1$’, if $h_Z$ becomes a subfunctor of its usual functor of points $h_Y$ (again, assigning to a monoid its integral monoid algebra $\mathbb{Z} M$) and $Y$ is the ‘best’ integral scheme approximation of the complex evaluation map.

To illustrate this, consider the painting Study after Velázquez’s Portrait of Pope Innocent X by Francis Bacon (right-hand painting above) which is a distorded version of the left-hand painting Portrait of Innocent X by Diego Velázquez.

Here, Velázquez’ painting plays the role of the complex variety which makes the combinatorial gadget $h_Z$ visible, and, Bacon’s painting depicts the integral scheme, build up from this combinatorial data, which approximates the evaluation map best.

All of the former approaches more or less give the same very small list of integral schemes defined over $\mathbb{F}_1$, none of them motivically interesting.

The current approach : Jackson Pollock “No. 8″ (1949)

An entirely different approach was proposed by James Borger in $\Lambda$-rings and the field with one element. He proposes another definition for commutative $\mathbb{F}_1$-algebras, namely $\lambda$-rings (in the sense of Grothendieck’s Riemann-Roch) and he argues that the $\lambda$-ring structure (which amounts in the sensible cases to a family of endomorphisms of the integral ring lifting the Frobenius morphisms) can be viewed as descent data from $\mathbb{Z}$ to $\mathbb{F}_1$.

The list of integral schemes of finite type with a $\lambda$-structure coincides roughly with the list of integral schemes defined over $\mathbb{F}_1$ in the other approaches, but Borger’s theory really shines in that it proposes long sought for mystery-objects such as $\mathbf{spec}(\mathbb{Z}) \times_{\mathbf{spec}(\mathbb{F}_1)} \mathbf{spec}(\mathbb{Z})$. If one accepts Borger’s premise, then this object should be the geometric object corresponding to the Witt-ring $W(\mathbb{Z})$. Recall that the role of Witt-rings in $\mathbb{F}_1$-geometry was anticipated by Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$.

But, Witt-rings and their associated Witt-spaces are huge objects, so one needs to extend arithmetic geometry drastically to include such ‘integral schemes of infinite type’. Borger has made a couple of steps in this direction in The basic geometry of Witt vectors, II: Spaces.

To depict these new infinite dimensional geometric objects I’ve chosen for Jackson Pollock‘s painting No. 8. It is no coincidence that Pollock-paintings also appeared in the depiction of noncommutative spaces. In fact, Matilde Marcolli has made the connection between $\lambda$-rings and noncommutative geometry in Cyclotomy and endomotives by showing that the Bost-Connes endomotives are universal for $\lambda$-rings.

Last time we did recall Manin’s comparisons between some approaches to geometry over the absolute point $\pmb{spec}(\mathbb{F}_1)$ and trends in the history of art.

In the comments to that post, Javier Lopez-Pena wrote that he and Oliver Lorscheid briefly contemplated the idea of extending Manin’s artsy-dictionary to all approaches they did draw on their Map of $\mathbb{F}_1$-land.

So this time, we will include here Javier’s and Oliver’s insights on the colored pieces below in their map : CC=Connes-Consani, Generalized torified schemes=Lopez Pena-Lorscheid, Generalized schemes with 0=Durov and, this time, $\Lambda$=Manin-Marcolli.

Durov : romanticism

In his 568 page long Ph.D. thesis New Approach to Arakelov Geometry Nikolai Durov introduces a vast generalization of classical algebraic geometry in which both Arakelov geometry and a more exotic geometry over $\mathbb{F}_1$ fit naturally. Because there were great hopes and expectations it would lead to a big extension of algebraic geometry, Javier and Oliver associate this approach to romantism. From wikipedia : “The modern sense of a romantic character may be expressed in Byronic ideals of a gifted, perhaps misunderstood loner, creatively following the dictates of his inspiration rather than the standard ways of contemporary society.”

Manin and Marcolli : impressionism

Yuri I. Manin in Cyclotomy and analytic geometry over $\mathbb{F}_1$ and Matilde Marcolli in Cyclotomy and endomotives develop a theory of analytic geometry over $\mathbb{F}_1$ based on analytic functions ‘leaking out of roots of unity’. Javier and Oliver depict such functions as ‘thin, but visible brush strokes at roots of 1′ and therefore associate this approach to impressionism. Frow wikipedia : ‘Characteristics of Impressionist paintings include: relatively small, thin, yet visible brush strokes; open composition; emphasis on accurate depiction of light in its changing qualities (often accentuating the effects of the passage of time); common, ordinary subject matter; the inclusion of movement as a crucial element of human perception and experience; and unusual visual angles.’

Connes and Consani : cubism

In On the notion of geometry over $\mathbb{F}_1$ Alain Connes and Katia Consani develop their extension of Soule’s approach. A while ago I’ve done a couple of posts on this here, here and here. Javier and Oliver associate this approach to cubism (a.o. Pablo Picasso and Georges Braque) because of the weird juxtapositions of the simple monoidal pieces in this approach.

Lopez-Pena and Lorscheid : deconstructivism

Torified varieties and schemes were introduced by Javier Lopez-Pena and Oliver Lorscheid in Torified varieties and their geometries over $\mathbb{F}_1$ to get lots of examples of varieties over the absolute point in the sense of both Soule and Connes-Consani. Because they were fragmenting schemes into their “fundamental pieces” they associate their approach to deconstructivism.

Another time I’ll sketch my own arty-farty take on all this.

This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program.

Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative boundary to the Langlands program (at least for $GL_1 $ and $GL_2 $ over the rationals $\mathbb{Q} $).

Recall that Langlands for $GL_1 $ over the rationals is the correspondence, given by the Artin reciprocity law, between on the one hand the abelianized absolute Galois group

$Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} = Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q}) \simeq \hat{\mathbb{Z}}^* $

and on the other hand the connected components of the idele classes

$\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast} = \mathbb{R}^{\ast}_{+} \times \hat{\mathbb{Z}}^{\ast} $

The locally compact Abelian group of idele classes can be viewed as the nice locus of the horrible quotient space of adele classes $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $. There is a well-defined map

$\mathbb{A}_{\mathbb{Q}}’/\mathbb{Q}^{\ast} \rightarrow \mathbb{R}_{+} \qquad (x_{\infty},x_2,x_3,\ldots) \mapsto | x_{\infty} | \prod | x_p |_p $

from the subset $\mathbb{A}_{\mathbb{Q}}’ $ consisting of adeles of which almost all terms belong to $\mathbb{Z}_p^{\ast} $. The inverse image of this map over $\mathbb{R}_+^{\ast} $ are precisely the idele classes $\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $. In this way one can view the adele classes as a closure, or ‘compactification’, of the idele classes.

This is somewhat reminiscent of extending the nice action of the modular group on the upper-half plane to its badly behaved action on the boundary as in the Manin-Marcolli cave post.

The topological properties of the fiber over zero, and indeed of the total space of adele classes, are horrible in the sense that the discrete group $\mathbb{Q}^* $ acts ergodically on it, due to the irrationality of $log(p_1)/log(p_2) $ for primes $p_i $. All this is explained well (in the semi-local case, that is using $\mathbb{A}_Q’ $ above) in the Connes-Marcolli book (section 2.7).

In much the same spirit as non-free actions of reductive groups on algebraic varieties are best handled using stacks, such ergodic actions are best handled by the tools of noncommutative geometry. That is, one tries to get at the geometry of $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $ by studying an associated non-commutative algebra, the skew-ring extension of the group-ring of the adeles by the action of $\mathbb{Q}^* $ on it. This algebra is known to be Morita equivalent to the Bost-Connes algebra which is the algebra featuring in Connes’ approach to the Riemann hypothesis.

It shouldn’t thus come as a major surprise that one is able to recover the other side of the Langlands correspondence, that is the Galois group $Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q}) $, from the Bost-Connes algebra as the symmetries of certain states.

In a similar vein one can read the Connes-Marcolli $GL_2 $-system (section 3.7 of their book) as an exploration of the noncommutative closure of the Langlands-space $GL_2(\mathbb{A}_{\mathbb{Q}})/GL_2(\mathbb{Q}) $.

At the moment I’m running a master-seminar noncommutative geometry trying to explain this connection in detail. But, we’re still in the early phases, struggling with the topology of ideles and adeles, reciprocity laws, L-functions and the lot. Still, if someone is interested I might attempt to post some lecture notes here.

In the wake of a colleague’s suicide and the suicide of three students, Matilde Marcolli gave an interesting and courageous talk at Caltech in April : The dark heart of our brightness: bipolar disorder and scientific creativity. Although these slides give a pretty good picture of the talk, if you can please take the time to watch it (the talk starts 44 minutes into the video).

Courageous because as the talk progresses, she gives more and more examples from her own experiences, thereby breaking the taboo surrounding the topic of bipolar mood disorder among scientists. Interesting because she raises a couple of valid points, well worth repeating.

We didn’t can see it coming

We are always baffled when someone we know commits suicide, especially if that person is extremely successful in his/her work. ‘(S)he was so full of activity!’, ‘We did not see it coming!’ etc. etc.

Matilde argues that if a person suffers from bipolar mood disorder (from mild forms to full-blown manic-depression), a condition quite common among scientists and certainly mathematicians, we can see it coming, if we look for the proper signals!

We, active scientists, are pretty good at hiding a down-period. We have collected an arsenal of tricks not to send off signals when we feel depressed, simply because it’s not considered cool behavior. On the other hand, in our manic phases, we are quite transparent because we like to show off our activity and creativity!

Matilde tells us to watch out for people behaving orders-of-magnitude out of their normal-mode behavior. Say, someone who normally posts one or two papers a year on the arXiv, suddenly posting 5 papers in one month. Or, someone going rarely to a conference, now spending a summer flying from one conference to the next. Or, someone not blogging for months, suddenly flooding you with new posts…

As scientists we are good at spotting such order-of-magnitude-out-behavior. So we can detect friends and colleagues going through a manic-phase and hence should always take such a person serious (and try to offer help) when they send out signals of distress.

Mood disorder, a Faustian bargain

The Faust legend :
“Despite his scholarly eminence, Faust is bored and disappointed. He decides to call on the Devil for further knowledge and magic powers with which to indulge all the pleasures of the world. In response, the Devil’s representative Mephistopheles appears. He makes a bargain with Faust: Mephistopheles will serve Faust with his magic powers for a term of years, but at the end of the term, the Devil will claim Faust’s soul and Faust will be eternally damned.”

Mathematicians suffering from mood disorder seldom see their condition as a menace, but rather as an advantage. They know they do their best and most creative work in short spells of intense activity during their manic phase and take the down-phase merely as a side effect. We fear that if we seek treatment, we may as well loose our creativity.

That is, like Faust, we indulge the pleasures of our magic powers during a manic-phase, knowing only too well that the devilish depression-phase may one day claim our life or mental sanity…

I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting

and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own post) let me point out that I am well aware of the controversy surrounding this painting.

This painting is among 32 works recently discovered and initially attributed to Pollock.
In fact, I’ve already told part of the story in Doodles worth millions (or not)? (thanks to PD1). The story involves the people on the right : from left to right, Jackson Pollock, his wife Lee Krasner, Mercedes Matter and her son Alex Matter.

Alex Matter, whose father, Herbert, and mother, Mercedes, were artists and friends of Jackson Pollock, discovered after his mother died a group of small drip paintings in a storage locker in Wainscott, N.Y. which he believed to be authentic Pollocks.

Read the post mentioned above if you want to know how mathematics screwed up his plan, or much better, reed the article Anatomy of the Jackson Pollock controversy by Stephen Litt.

So, perhaps the painting above was not the smartest choice, but we could take any other genuine Pollock ‘drip-painting’, a technique he taught himself towards the end of 1946 to make an image by splashing, pouring, sloshing colors onto the canvas. Typically, such a painting consists of blops of paint, connected via thin drip-lines.

What does this have to do with noncommutative geometry? Well, consider the blops as ‘points’. In commutative geometry, distinct points cannot share tangent information¹. In the noncommutative world though, they can!, or if you want to phrase it like this, noncommutative points ‘can talk to each other’. And, that’s what we cherish in those drip-lines.

But then, if two points share common tangent informations, they must be awfully close to each other… so one might imagine these Pollock-lines to be strings holding these points together. Hence, it would make more sense to consider the ‘Pollock-quotient-painting’, that is, the space one gets after dividing out the relation ‘connected by drip-lines’².

For this reason, my own mental picture of a genuinely noncommutative space ³ looks more like the picture below

The colored blops you see are really sets of points which you might view as, say, a FacebookGroup⁴. Some chatter may occur between two distinct FacebookGroups, the more chatter the thicker the connection depicted⁵. Now, there are some tiny isolated spots (say blue ones in the upper right-hand quadrant). These should really be looked at as remote clusters of noncommutative points (sharing no (tangent) information whatsoever with the blops in the foregound). If we would zoom into them beyond the Planck scale (if I’m allowed to say a bollock-word in a Pollock-post) they might reveal again a whole universe similar to the interconnected blops upfront.

The picture was produced using the fabulous Pollock engine. Just use your mouse to draw and click to change colors in order to produce your very own noncommutative space!

For the mathematicians still around, this may sound like a lot of Pollock-bollocks but can be made precise. See my note Noncommutative geometry and dual coalgebras for a very terse reading. Now that coalgebras are gaining popularity, I really should write a more readable account of it, including some fanshi-wanshi examples…

1. technically : a commutative semi-local ring splits as the direct sum of local rings and this does no longer hold for a noncommutative semi-local ring
2. my guess is that Matilde thinks of the lines as the action of a group on the points giving a topological horrible quotient space, and thats precisely where noncommutative geometry shines
3. that is, the variety corresponding to a huge noncommutative algebra such as free algebras, group algebras of arithmetic groups or fundamental groups
4. technically, think of them as the connected components of isomorphism classes of finite dimensional simple representations of your favorite noncommutative algebra
5. technically, the size of the connection is the dimension of the ext-group between generic simples in the components

The general public expects pictures from geometers, even from non-commutative geometers. Hence, it is important for researchers in this topic to make an attempt to convey the mental picture they have of their favourite noncommutative space, … somehow. Two examples :

This picture was created by Shahn Majid. It appears on his visions of noncommutative geometry page as well as in an extremely readable Plus-magazine article on Quantum geometry, written by Marianne Freiberger, explaining Shahn’s ideas. For more information on this, read Shahn’s SpaceTime blog.

This painting is Jackson Pollock‘s “Untitled N.3″. It depicts the way Matilde Marcolli imagines a noncommutative space. It is taken from her slides of her talk for a general audience Mathematicians look at particle physics.

Of course, excellent math-blogs exist in every language imaginable, but my linguistic limitations restrict me to the ones written in English, French, German and … Dutch. Here a few links to Dutch (or rather, Flemish) math-blogs, in order of proximity :
Stijn Symens blog, Rudy Penne’s wiskunde is sexy (math is sexy), Koen Vervloesem’s QED.

My favorite one is wiskundemeisjes (‘math-chicks’ or ‘math-girls’), written by Ionica Smeets and Jeanine Daems, two reasearchers at Leiden University. Every month they have a post called “the favorite (living) mathematician of …” in which they ask someone to nominate and introduce his/her favorite colleague mathematician. Here some examples : Roger Penrose chooses Michael Atiyah, Robbert Dijkgraaf chooses Maxim Kontsevich, Frans Oort chooses David Mumford, Gunther Cornelissen chooses Yuri I. Manin, Hendrik Lenstra chooses Bjorn Poonen, etc. the full list is here or here. This series deserves a wider audience. Perhaps Ionica and Jeanine might consider translating some of these posts?

I’m certain their English is far better than mine, so here’s a feeble attempt to translate the one post in their series they consider a complete failure (it isn’t even listed in the category). Two reasons for me to do so : it features Matilde Marcolli (one of my own favorite living mathematicians) and Matilde expresses here very clearly my own take on popular-math books/blogs.

The original post was written by Ionica and was called Weg met de ‘favoriete wiskundige van…’ :

“This week I did spend much of my time at the Fifth European Mathematical Congress in Amsterdam. Several mathematicians suggested I should have a chat with Matilde Marcolli, one of the plenary speakers. It seemed like a nice idea to ask her about her favorite (still living) mathematician, for our series.

Marcolli explained why she couldn’t answer this question : she has favorite mathematical ideas, but it doesn’t interest her one bit who discovered or proved them. And, there are mathematicians she likes, but that’s because she finds them interesting as human beings, independent of their mathematical achievements.

In addition, she thinks it’s a mistake to focus science too much on the persons. Scientific ideas should play the main role, not the scientists themselves. To her it is important to remember that many results are the combined effort of several people, that science doesn’t evolve around personalities and that scientific ideas are accessible to anyone.

Marcolli also dislikes the current trend in popular science writing: “I am completely unable to read popular-scientific books. As soon as they start telling anecdotes and stories, I throw away the book. I don’t care about their lives, I care about the real stuff.”

She’d love to read a popular science-book containing only ideas. She regrets that most of these books restrict to story-telling, but fail to disseminate the scientific ideas.”

Ionica then goes on to defend her own approach to science-popularization :

“… Probably, people will not know much about Galois-theory by reading about his turbulent life. Still, I can imagine people to become interested in ‘the real stuff’ after reading his biography, and, in this manner they will read some mathematics they wouldn’t have known to exist otherwise. But, Marcolli got me thinking, for it is true that almost all popular science-books focus on anecdotes rather than science itself. Is this wrong? For instance, do you want to see more mathematics here? I’m curious to hear your opinion on this.”

Even though my own approach is somewhat different, Ionica and Jeanine you’re doing an excellent job: “houden zo!”

At the Max-Planck Institute in Bonn Yuri Manin gave a talk about the field of one element, $\mathbb{F}_1 $ earlier this week entitled “Algebraic and analytic geometry over the field F_1″.

Moreover, Javier Lopez-Pena and Bram Mesland will organize a weekly “F_un Study Seminar” starting next tuesday.

Over at Noncommutative Geometry there is an Update on the field with one element pointing us to a YouTube-clip featuring Alain Connes explaining his paper with Katia Consani and Matilde Marcolli entitled “Fun with F_un”. Here’s the clip

Finally, as I’ll be running a seminar here too on F_un, we’ve set up a group blog with the people from MPI (clearly, if you are interested to join us, just tell!). At the moment there are just a few of my old F_un posts and a library of F_un papers, but hopefully a lot will be added soon. So, have a look at F_un mathematics

There are only a handful of human activities where one goes to extraordinary lengths to keep a dream alive, in spite of overwhelming evidence : religion, theoretical physics, supporting the Belgian football team and … mathematics.

In recent years several people spend a lot of energy looking for properties of an elusive object : the field with one element $\mathbb{F}_1 $, or in French : “F-un”. The topic must have reached a level of maturity as there was a conference dedicated entirely to it : NONCOMMUTATIVE GEOMETRY AND GEOMETRY OVER THE FIELD WITH ONE ELEMENT.

In this series I’d like to find out what the fuss is all about, why people would like it to exist and what it has to do with noncommutative geometry. However, before we start two remarks :

The field $\mathbb{F}_1 $ does not exist, so don’t try to make sense of sentences such as “The ‘field with one element’ is the free algebraic monad generated by one constant (p.26), or the universal generalized ring with zero (p.33)” in the wikipedia-entry. The simplest proof is that in any (unitary) ring we have $0 \not= 1 $ so any ring must contain at least two elements. A more highbrow version : the ring of integers $\mathbb{Z} $ is the initial object in the category of unitary rings, so it cannot be an algebra over anything else.

The second remark is that several people have already written blog-posts about $\mathbb{F}_1 $. Here are a few I know of : David Corfield at the n-category cafe and at his old blog, Noah Snyder at the secret blogging seminar, Kea at the Arcadian functor, AC and K. Consani at Noncommutative geometry and John Baez wrote about it in his weekly finds.

The dream we like to keep alive is that we will prove the Riemann hypothesis one fine day by lifting Weil’s proof of it in the case of curves over finite fields to rings of integers.

Even if you don’t know a word about Weil’s method, if you think about it for a couple of minutes, there are two immediate formidable problems with this strategy.

For most people this would be evidence enough to discard the approach, but, we mathematicians have found extremely clever ways for going into denial.

The first problem is that if we want to think of $\mathbf{spec}(\mathbb{Z}) $ (or rather its completion adding the infinite place) as a curve over some field, then $\mathbb{Z} $ must be an algebra over this field. However, no such field can exist…

No problem! If there is no such field, let us invent one, and call it $\mathbb{F}_1 $. But, it is a bit hard to do geometry over an illusory field. Christophe Soule succeeded in defining varieties over $\mathbb{F}_1 $ in a talk at the 1999 Arbeitstagung and in a more recent write-up of it : Les varietes sur le corps a un element.

We will come back to this in more detail later, but for now, here’s the main idea. Consider an existent field $k $ and an algebra $k \rightarrow R $ over it. Now study the properties of the functor (extension of scalars) from $k $-schemes to $R $-schemes. Even if there is no morphism $\mathbb{F}_1 \rightarrow \mathbb{Z} $, let us assume it exists and define $\mathbb{F}_1 $-varieties by requiring that these guys should satisfy the properties found before for extension of scalars on schemes defined over a field by going to schemes over an algebra (in this case, $\mathbb{Z} $-schemes). Roughly speaking this defines $\mathbb{F}_1 $-schemes as subsets of points of suitable $\mathbb{Z} $-schemes.

But, this is just one half of the story. He adds to such an $\mathbb{F}_1 $-variety extra topological data ‘at infinity’, an idea he attributes to J.-B. Bost. This added feature is a $\mathbb{C} $-algebra $\mathcal{A}_X $, which does not necessarily have to be commutative. He only writes : “Par ignorance, nous resterons tres evasifs sur les proprietes requises sur cette $\mathbb{C} $-algebre.”

The algebra $\mathcal{A}_X $ originates from trying to bypass the second major obstacle with the Weil-Riemann-strategy. On a smooth projective curve all points look similar as is clear for example by noting that the completions of all local rings are isomorphic to the formal power series $k[[x]] $ over the basefield, in particular there is no distinction between ‘finite’ points and those lying at ‘infinity’.

The completions of the local rings of points in $\mathbf{spec}(\mathbb{Z}) $ on the other hand are completely different, for example, they have residue fields of different characteristics… Still, local class field theory asserts that their quotient fields have several common features. For example, their Brauer groups are all isomorphic to $\mathbb{Q}/\mathbb{Z} $. However, as $Br(\mathbb{R}) = \mathbb{Z}/2\mathbb{Z} $ and $Br(\mathbb{C}) = 0 $, even then there would be a clear distinction between the finite primes and the place at infinity…

Alain Connes came up with an extremely elegant solution to bypass this problem in Noncommutative geometry and the Riemann zeta function. He proposes to replace finite dimensional central simple algebras in the definition of the Brauer group by AF (for Approximately Finite dimensional)-central simple algebras over $\mathbb{C} $. This is the origin and the importance of the Bost-Connes algebra.

We will come back to most of this in more detail later, but for the impatient, Connes has written a paper together with Caterina Consani and Matilde Marcolli Fun with $\mathbb{F}_1 $ relating the Bost-Connes algebra to the field with one element.

By now, everyone remotely interested in Connes’ approach to the Riemann hypothesis, knows the _one line mantra_

one can use noncommutative geometry to extend Weil’s proof of the Riemann-hypothesis in the function field case to that of number fields

But, can one go beyond this sound-bite in a series of blog posts? A few days ago, I was rather optimistic, but now, after reading-up on the Connes-Consani-Marcolli project, I feel overwhelmed by the sheer volume of their work (and by my own ignorance of key tools in the approach). The most recent account takes up half of the 700+ pages of the book Noncommutative Geometry, Quantum Fields and Motives by Alain Connes and Matilde Marcolli…

So let us set a more modest goal and try to understand one of the first papers Alain Connes wrote about the RH : Noncommutative geometry and the Riemann zeta function. It is only 24 pages long and relatively readable. But even then, the reader needs to know about class field theory, the classification of AF-algebras, Hecke algebras, etc. etc. Most of these theories take a book to explain. For example, the first result he mentions is the main result of local class field theory which appears only towards the end of the 200+ pages of Jean-Pierre Serre’s Local Fields, itself a somewhat harder read than the average blogpost…

Anyway, we will see how far we can get. Here’s the plan : I’ll take the heart-bit of their approach : the Bost-Connes system, and will try to understand it from an algebraist’s viewpoint. Today we will introduce the groups involved and describe their cosets.

For any commutative ring $R $ let us consider the group of triangular $2 \times 2 $ matrices of the form

$P_R = { \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix}~|~b \in R, a \in R^* } $

(that is, $a $ in an invertible element in the ring $R $). This is really an affine group scheme defined over the integers, that is, the coordinate ring

$\mathbb{Z}[P] = \mathbb{Z}[x,x^{-1},y] $ becomes a Hopf algebra with comultiplication encoding the group-multiplication. Because

$\begin{bmatrix} 1 & b_1 \\ 0 & a_1 \end{bmatrix} \begin{bmatrix} 1 & b_2 \\ 0 & a_2 \end{bmatrix} = \begin{bmatrix} 1 & 1 \times b_2 + b_1 \times a_2 \\ 0 & a_1 \times a_2 \end{bmatrix} $

we have $\Delta(x) = x \otimes x $ and $\Delta(y) = 1 \otimes y + y \otimes x $, or $x $ is a group-like element whereas $y $ is a skew-primitive. If $R \subset \mathbb{R} $ is a subring of the real numbers, we denote by $P_R^+ $ the subgroup of $P_R $ consisting of all matrices with $a > 0 $. For example,

$\Gamma_0 = P_{\mathbb{Z}}^+ = { \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}~|~n \in \mathbb{Z} } $

which is a subgroup of $\Gamma = P_{\mathbb{Q}}^+ $ and our first job is to describe the cosets.

The left cosets $\Gamma / \Gamma_0 $ are the subsets $\gamma \Gamma_0 $ with $\gamma \in \Gamma $. But,

$\begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & b+n \\ 0 & a \end{bmatrix} $

so if we represent the matrix $\gamma = \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} $ by the point $~(a,b) $ in the right halfplane, then for a given positive rational number $a $ the different cosets are represented by all $b \in [0,1) \cap \mathbb{Q} = \mathbb{Q}/\mathbb{Z} $. Hence, the left cosets are all the rational points in the region between the red and green horizontal lines. For fixed $a $ the cosets correspond to the rational points in the green interval (such as over $\frac{2}{3} $ in the picture on the left.

Similarly, the right cosets $\Gamma_0 \backslash \Gamma $ are the subsets $\Gamma_0 \gamma $ and as

$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} = \begin{bmatrix} 1 & b+na \\ 0 & a \end{bmatrix} $

we see similarly that the different cosets are precisely the rational points in the region between the lower red horizontal and the blue diagonal line. So, for fixed $a $ they correspond to rational points in the blue interval (such as over $\frac{3}{2} $) $[0,a) \cap \mathbb{Q} $. But now, let us look at the double coset space $\Gamma_0 \backslash \Gamma / \Gamma_0 $. That is, we want to study the orbits of the action of $\Gamma_0 $, acting on the right, on the left-cosets $\Gamma / \Gamma_0 $, or equivalently, of the action of $\Gamma_0 $ acting on the left on the right-cosets $\Gamma_0 \backslash \Gamma $. The crucial observation to make is that these actions have finite orbits, or equivalently, that $\Gamma_0 $ is an almost normal subgroup of $\Gamma $ meaning that $\Gamma_0 \cap \gamma \Gamma_0 \gamma^{-1} $ has finite index in $\Gamma_0 $ for all $\gamma \in \Gamma $. This follows from

$\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & b \\ 0 & a \end{bmatrix} \begin{bmatrix} 1 & m \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & b+m+an \\ 0 & a \end{bmatrix} $

and if $n $ varies then $an $ takes only finitely many values modulo $\mathbb{Z} $ and their number depends only on the denominator of $a $. In the picture above, the blue dots lying on the line over $\frac{2}{3} $ represent the double coset

$\Gamma_0 \begin{bmatrix} 1 & \frac{2}{3} \\ 0 & \frac{2}{3} \end{bmatrix} $ and we see that these dots split the left-cosets with fixed value $a=\frac{2}{3} $ (that is, the green line-segment) into three chunks (3 being the denominator of a) and split the right-cosets (the line-segment under the blue diagonal) into two subsegments (2 being the numerator of a). Similarly, the blue dots on the line over $\frac{3}{2} $ divide the left-cosets in two parts and the right cosets into three parts.

This shows that the $\Gamma_0 $-orbits of the right action on the left cosets $\Gamma/\Gamma_0 $ for each matrix $\gamma \in \Gamma $ with $a=\frac{2}{3} $ consist of exactly three points, and we denote this by writing $L(\gamma) = 3 $. Similarly, all $\Gamma_0 $-orbits of the left action on the right cosets $\Gamma_0 \backslash \Gamma $ with this value of a consist of two points, and we write this as $R(\gamma) = 2 $.

For example, on the above picture, the black dots on the line over $\frac{2}{3} $ give the matrices in the double coset of the matrix

$\gamma = \begin{bmatrix} 1 & \frac{1}{7} \\ 0 & \frac{2}{3} \end{bmatrix} $

and the gray dots on the line over $\frac{3}{2} $ determine the elements of the double coset of

$\gamma^{-1} = \begin{bmatrix} 1 & -\frac{3}{14} \\ 0 & \frac{3}{2} \end{bmatrix} $

and one notices (in general) that $L(\gamma) = R(\gamma^{-1}) $. But then, the double cosets with $a=\frac{2}{3} $ are represented by the rational b’s in the interval $[0,\frac{1}{3}) $ and those with $a=\frac{3}{2} $ by the rational b’s in the interval $\frac{1}{2} $. In general, the double cosets of matrices with fixed $a = \frac{r}{s} $ with $~(r,s)=1 $ are the rational points in the line-segment over $a $ with $b \in [0,\frac{1}{s}) $.

That is, the Bost-Connes double coset space $\Gamma_0 \backslash \Gamma / \Gamma_0 $ are the rational points in a horrible fractal comb. Below we have drawn only the part of the dyadic values, that is when $a = \frac{r}{2^t} $ in the unit inverval

and of course we have to super-impose on it similar pictures for rationals with other powers as their denominators. Fortunately, NCG excels in describing such fractal beasts…

UPDATE : here is a slightly beter picture of the coset space, drawing the part over all rational numbers contained in the 15-th Farey sequence. The blue segments of length one are at 1,2,3,…