Since the last activity here, there have been some interesting new Fun-developments, Connes and Consani posted at least two papers, Marcolli at least one, Manin put a new version of his cyclotomy paper online and there was even a ‘survey’-paper relating all different notions of Fun-geometry. So, imho, there’s plenty of new material to cover to keep the site going for a while. Interested in helping out?

I will try to convince you that Grothendieck’s ‘dessins d’enfant’ form an example of a noncommutative manifold over the mythical field with one element (in the sense of Soule and Connes-Consani).

However, dessins only appear at the final slide. The main part of the talk consisted in explaining one sentence in Manin’s recent paper (page 4, line 3):

Soule’s definition of an -scheme involves besides , a -algebra , and each cyclotomic point of coming from must assign ‘values’ to the elements of . His choice of for the multiplicative group is that of continuous functions on the unit circle in … We suggest to consider the ring of Habiro’s analytic functions…

I promised Jacques to do a proper write-up of the talk (and include some more details on the final slide) so I might as well do a couple of posts on it, later.

After a very brief historical intro, I’ll focus on this basic question : what geometric object is considered over the ‘absolute point’ ? In particular, what is its ‘dimension’ and what geometric object corresponds to prime numbers? I’ll follow the answers and motivations given by Yuri Manin in The notion of dimension in geometry and algebra. There are (at least) three different answers to these questions :

dim = 1 : This is the classical approach we know from global field theory, is analogous to the affine line over a finite field, and, more generally prime ideals in number fields correspond to points on curves (over finite fields). Applications are to the Riemann hypothesis on zeta functions using the concept of ‘absolute motives’ as in Manin’s 1995 paper and recent work of Kurokawa.

dim = 3 : This is based on the Artin-Verdier-Mazur duality in etale topology suggesting that might be considered as the three-sphere with prime ideals corresponding to knots. Applications include the interpretation of power residue symbols and reciprocity laws as (higher) linking numbers as in the Kapranov-Smirnov paper and supported by recent work of Morishita.

dim = : This is supported by the fact that we are unable to realize as an affine -variety. Still, in his recent paper, Manin suggests that a Soule-version of Witt vectors, by restricting the values of the ‘ghost variables’ to cyclotomic numbers, might furnish a formal -approximation to the elusive arithmetic line. In this set-up, primes correspond to factors of these Witt-functors as exemplified by the decomposition of the profinite numbers.

My prep-notes are far from ideal and are only meant to prevent me from getting too lost in these ghost-worlds. Anyway, here they are. Comments are very wellcome!

I hope to turn these notes into a series of more readable posts in the upcoming days and weeks…

It is a bit odd hearing a talk without seeing any slides or blackboard images, but then, here is the streaming audio (you need to have Real Player installed).

Consani F_un talk at Fields Institute

which over resp. become visible by a complex (resp. integral) algebra having suitable universal properties.

However, we didn’t specify what we meant by a complex noncommutative variety (resp. an integral noncommutative scheme). In particular, we claimed that the -’points’ associated to the functor

(here denotes all elements of order of )

were precisely the modular dessins d’enfants of Grothendieck, but didn’t give details. We’ll try to do this now.

For algebras over a field we follow the definition, due to Kontsevich and Soibelman, of so called “noncommutative thin schemes”. Actually, the thinness-condition is implicit in both Soule’s-approach as that of Connes and Consani : we do not consider R-points in general, but only those of rings R which are finite and flat over our basering (or field).

So, what is a noncommutative thin scheme anyway? Well, its a covariant functor (commuting with finite projective limits)

from finite-dimensional (possibly noncommutative) -algebras to sets. Now, the usual dual-space operator gives an anti-equivalence of categories

so a thin scheme can also be viewed as a contra-variant functor (commuting with finite direct limits)

In particular, we are interested to associated to any {tex]k[/tex]-algebra its representation functor :

This may look strange at first sight, but is a finite dimensional algebra and any -dimensional representation of is an algebra map and we take to be the dual coalgebra of this image.

Kontsevich and Soibelman proved that every noncommutative thin scheme is representable by a -coalgebra. That is, there exists a unique coalgebra (which they call the coalgebra of ‘distributions’ of ) such that for every finite dimensional -algebra we have

In the case of interest to us, that is for the functor the coalgebra of distributions is Kostant’s dual coalgebra . This is the not the full linear dual of but contains only those linear functionals on which factor through a finite dimensional quotient.

So? You’ve exchanged an algebra for some coalgebra , but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose is the coordinate ring of a smooth affine variety , then its dual coalgebra looks like

the direct sum of all universal (co)algebras of tangent spaces at points . But how do we get the variety out of this? Well, any coalgebra has a coradical (being the sun of all simple subcoalgebras) and in the case just mentioned we have

so every point corresponds to a unique simple component of the coradical. In the general case, the coradical of the dual coalgebra is the direct sum of all simple finite dimensional representations of . That is, the direct summands of the coalgebra give us a noncommutative variety whose points are the simple representations, and the remainder of the coalgebra of distributions accounts for infinitesimal information on these points (as do the tangent spaces in the commutative case).

In fact, it was a surprise to me that one can describe the dual coalgebra quite explicitly, and that -structures make their appearance quite naturally. See this paper if you’re in for the details on this.

That settles the problem of what we mean by the noncommutative variety associated to a complex algebra. But what about the integral case? In the above, we used extensively the theory of Kostant-duality which works only for algebras over fields…

Well, not quite. In the case of (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual -coalgebra of an algebra (which is -torsion free)

(over general rings there may be also variants of this duality, as in Street’s book an Quantum groups). Probably lots of people have come up with this, but the only explicit reference I have is to the first paper I’ve ever written. So, also for algebras over we can define a suitable noncommutative integral scheme (the coradical approach accounts only for the maximal ideals rather than all primes, but somehow this is implicit in all approaches as we consider only thin schemes).

Fine! So, we can make sense of the noncommutative geometrical objects corresponding to the group-algebras and where is the modular group (the algebras corresponding to the -functor). But, what might be the points of the noncommutative scheme corresponding to ???

Well, let’s continue the path cut out before. “Points” should correspond to finite dimensional “simple representations”. Hence, what are the finite dimensional simple -representations of ? (Or, for that matter, of any group )

Here we come back to Javier’s post on this : a finite dimensional -vectorspace is a finite set. A -representation on this set (of n-elements) is a group-morphism

hence it gives a permutation representation of on this set. But then, if finite dimensional -representations of are the finite permutation representations, then the simple ones are the transitive permutation representations. That is, the points of the noncommutative scheme corresponding to are the conjugacy classes of subgroups such that is finite. But these are exactly the modular dessins d’enfants introduced by Grothendieck as I explained a while back elsewhere (see for example this post and others in the same series).

It is far from easy to read (at least for me), but I’ve found every effort trying to make sense of even a single paragraph time well invested. Here are the opening lines :

“Analogies between number fields and function fields have been a long-time source of inspiration in arithmetic. However, one of the most intruiging problems in this approach, namely the problem of the absolute point, is still far from being satisfactorily understood.

The scheme , the final object in the category of schemes, has dimension 1 with respect to the Zariski topology and at least 3 with respect to the etale topology.

This generated a long-standing desire to introduce a more mythical object , the “absolute point” with a natural morphism

given for any arithmetic scheme so that global invariants of have an interpretation in terms of a version of direct image with respect to .”

And a few paragraphs further they issue a firm warning to all who think they can attack this problem without proper preparation.

“First of all, it is an old idea to interpret combinatorics of finite sets as the limit of linear algebra over the finite fields . This had lead to frequent consideration of the folklore object , the “field with one element”.

One can postulate, of course, that is the absolute point, but the real problem is to develop non-trivial consequences from this point of view.“

Perhaps, the -idea might have led to a real breakthrough if it had landed around 1930 in Germany. Whether it will do the same today, remains to be seen…

My guess of why they stop there is as good as anyone’s. Perhaps they felt that there is already enough noncommutativity in Soule’s gadget-approach (the algebra as in this post may very well be noncommutative). Perhaps they were only interested in the Bost-Connes system which can be entirely encoded in their commutative -geometry. Perhaps they felt unsure as to what the noncommutative scheme of an affine noncommutative algebra might be. Perhaps …

Remains the fact that their approach screams for a noncommutative extension. Their basic object is a covariant functor

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to . In a series of posts on the Connes-Consani paper (starting here) I took some care of getting rid of all scheme-lingo and rephrasing everything entirely into algebras. But then, this set-up can be extended verbatim to noncommuative -geometry, which should start from a covariant functor

from all finite groups to sets. Let’s recall quickly what the additional info should be making this functor a noncommutative (affine) F_un scheme :

There should be a finitely generated -algebra together with a natural transformation (the ‘evaluation’)

(both and the group-algebra may be noncommutative). The pair is then called a gadget and there is an obvious notion of ‘morphism’ between gadgets.

The crucial extra ingredient is an affine -algebra (possibly noncommutative) such that is a subfunctor of together with the following universal property :

any affine -algebra having a gadget-morphism comes from a -algebra morphism . (If this sounds too cryptic for you, please read the series on C-C mentioned before).

So, there is no problem in defining noncommutative affine F_un-schemes. However, as with any generalization, this only makes sense provided (a) we get something new and (b) we have interesting examples, not covered by the restricted theory.

At first sight we do not get something new as in the only example we did in the C-C-series (the forgetful functor) it is easy to prove (using the same proof as given in this post) that the forgetful-functor still has as its integral form the integral torus . However, both theories quickly diverge beyond this example.

For example, consider the functor

Then, if we restrict to abelian finite groups it is easy to see (again by a similar argument) that the two-dimensional integer torus is the correct integral form. However, this algebra cannot be the correct form for the functor on the category of all finite groups as any -algebra map determines (and is determined by) a pair of commuting units in , so the above functor can not be a subfunctor if we allow non-Abelian groups.

But then, perhaps there isn’t a minimal integral -form for this functor? Well, yes there is. Take the free group in two letters (that is, all words in noncommuting and satisfying only the trivial cancellation laws between a letter and its inverse), then the corresponding integral group-algebra does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map and we have to show, using the universal property that the image of is contained in the integral group-algebra . Take a generator of then the degree of the image is bounded say by and we can always find a subgroup such that is a fnite group and the quotient map is injective on the subspace spanned by all words of degree strictly less than . Then, the usual diagram-chase finishes the proof.

What makes this work is that the free group has ‘enough’ subgroups of finite index, a property it shares with many interesting discrete groups. Whence the blurb-message :

if the integers see a discrete group , then the field sees its profinite completion

So, yes, we get something new by extending the Connes-Consani approach to the noncommutative world, but do we have interesting examples? As “interesting” is a subjective qualification, we’d better invoke the authority-argument.

Alexander Grothendieck (sitting on the right, manifestly not disputing a vacant chair with Jean-Pierre Serre, drinking on the left (a marvelous picture taken by F. Hirzebruch in 1958)) was pushing the idea that profinite completions of arithmetical groups were useful in the study of the absolute Galois group , via his theory of dessins d’enfants (children;s drawings).

In a previous life, I’ve written a series of posts on dessins d’enfants, so I’ll restrict here to the basics. A smooth projective -curve has a Belyi-map ramified only in three points . The “drawing” corresponding to is a bipartite graph, drawn on the Riemann surface obtained by lifting the unit interval to . As the absolute Galois group acts on all such curves (and hence on their corresponding drawings), the action of it on these dessins d’enfants may give us a way into the multiple mysteries of the absolute Galois group.

In his “Esquisse d’un programme” (Sketch of a program if you prefer to read it in English) he writes :

“C’est ainsi que mon attention s’est portée vers ce que j’ai appelé depuis la “géométrie algêbrique anabélienne”, dont le point de départ est justement une étude (pour le moment limitée à la caractéristique zéro) de l’action de groupe de Galois “absolus” (notamment les groupes , ou est une extension de type fini du corps premier) sur des groupes fondamentaux géométriques (profinis) de variétés algébriques (définies sur ), et plus particulièrement (rompant avec une tradition bien enracinée) des groupes fondamentaux qui sont trés éloignés des groupes abéliens (et que pour cette raison je nomme “anabéliens”). Parmi ces groupes, et trés proche du groupe , il y a le compactifié profini du groupe modulaire , dont le quotient par le centre contient le précédent comme sous-groupe de congruence mod 2, et peut s’interpréter d’ailleurs comme groupe “cartographique” orienté, savoir celui qui classifie les cartes orientées triangulées (i.e. celles dont les faces des triangles ou des monogones).”

and a bit further, he writes :

“L’élément de structure de qui me fascine avant tout, est bien sur l’action extérieure du groupe de Galois sur le compactifié profini. Par le théorème de Bielyi, prenant les compactifiés profinis de sous-groupes d’indice fini de , et l’action extérieure induite (quitte à passer également à un sous-groupe overt de ), on trouve essentiellement les groupes fondamentaux de toutes les courbes algébriques définis sur des corps de nombres , et l’action extérieure de dessus.”

So, is there a noncommutative affine variety over of which the unique minimal integral model is the integral group algebra of the modular group (with ? Yes, here it is

where is the set of all elements of order in . The reason behind this is that the modular group is the free group product .

Fine, you may say, but all this is just algebra. Where is the noncommutative complex variety or the noncommutative integral scheme in all this? Well, we can introduce them too but as this post is already 1300 words long, I’ll better leave this for another time. In case you cannot stop thinking about it, here’s the short answer.

The complex noncommutative variety has as its ‘points’ all finite dimensional simple complex representations of the modular group, and the ‘points’ of the noncommutative -scheme are exactly the (modular) dessins d’enfants…

While this is a historic fact, and, Jacques Tits cannot be given enough credit for bringing a touch of surrealism into mathematics, but this is not the main drive for people getting into F_un, today.

There is a much deeper and older motivation behind most papers published recently on . Few of the authors will be willing to let you in on the secret, though, because if they did, it would sound much too presumptuous…

So, let’s have it out into the open : F_un mathematics’ goal is no less than proving the Riemann Hypothesis.

And even then, authors hide behind a smoke screen. The ‘official’ explanation being “we would like to copy Weil’s proof of the Riemann hypothesis in the case of function fields of curves over finite fields, by considering spec(Z) as a ‘curve’ over an algebra ‘dessous’ Z namely “. Alas, at this moment, none of the geometric approaches over the field with one element can make this stick.

Believe me for once, the main Jugendtraum of most authors is to get a grip on cyclotomy over . It is no accident that Connes makes a dramatic pauze in his YouTubeVideo to let the viewer see this equation on the backboard

But, what is the basis of all this childlike enthusiasm? A somewhat concealed clue is given in the introduction of the Kapranov-Smirnov paper. They write :

“In [?] the affine line over was considered; it consists formally of 0 and all the roots of unity. Put slightly differently, this leads to the consideration of “algebraic extensions” of . By analogy with genuine finite fields we would like to think that there is exactly one such extension of any given degree n, denote it by .

Of course, does not exist in a rigorous sense, but we can think if a scheme contains n-th roots of unity, then it is defined over , so that there is a morphism

The point of view that adjoining roots of unity is analogous to the extension of the base field goes back, at least to Weil (Lettre a Artin, Ouvres, vol 1) and Iwasawa…“

Okay, so rush down to your library, pick out the first of three volumes of Andre Weil’s collected works, look up his letter to Emil Artin written on July 10th 1942 (19 printed pages!), and head for the final section. Weil writes :

“Our proof of the Riemann hypothesis (in the function field case, red.) depended upon the extension of the function-fields by roots of unity, i.e. by constants; the way in which the Galois group of such extensions operates on the classes of divisors in the original field and its extensions gives a linear operator, the characteristic roots (i.e. the eigenvalues) of which are the roots of the zeta-function.

On a number field, the nearest we can get to this is by adjunction of -th roots of unity, being fixed; the Galois group of this infinite extension is cyclic, and defines a linear operator on the projective limit of the (absolute) class groups of those successive finite extensions; this should have something to do with the roots of the zeta-function of the field. However, our extensions are ramified (but only at a finite number of places, viz. the prime divisors of ). Thus a preliminary study of similar problems in function-fields might enable one to guess what will happen in number-fields.”

A few years later, in 1947, he makes this a bit more explicit in his marvelous essay “L’avenir des mathematiques” (The future of mathematics). Weil is still in shell-shock after the events of the second WW, and writes in beautiful archaic French sentences lasting forever :

“L’hypothèse de Riemann, après qu’on eut perdu l’espoir de la démontrer par les méthodes de la théorie des fonctions, nous apparaît aujourd’hui sous un jour nouveau, qui la montre inséparable de la conjecture d’Artin sur les fonctions L, ces deux problèmes étant deux aspects d’une même question arithmético-algébrique, où l’étude simultanée de toutes les extensions cyclotomiques d’un corps de nombres donné jouera sans doute le rôle décisif.

L’arithmétique gaussienne gravitait autour de la loi de réciprocité quadratique; nous savons maintenant que celle-ci n’est qu’un premier example, ou pour mieux dire le paradigme, des lois dites “du corps de classe”, qui gouvernent les extensions abéliennes des corps de nobres algébriques; nous savons formuler ces lois de manière à leur donner l’aspect d’un ensemble cohérent; mais, si plaisante à l’Å“il que soit cette façade, nous ne savons si elle ne masque pas des symmétries plus cachées.

Les automorphismes induits sur les groupes de classes par les automorphismes du corps, les propriétés des restes de normes dans les cas non cycliques, le passage à la limite (inductive ou projective) quand on remplace le corps de base par des extensions, par example cyclotomiques, de degré indéfiniment croissant, sont autant de questions sur lesquelles notre ignorance est à peu près complète, et dont l’étude contient peut-être la clef de l’hypothese de Riemann; étroitement liée à celles-ci est l’étude du conducteur d’Artin, et en particulier, dans le cas local, la recherche de la représentation dont la trace s’exprime au moyen des caractères simples avec des coefficients égaux aux exposants de leurs conducteurs.

Ce sont là quelques-unes des directions qu’on peut et qu’on doit songer à suivre afin de pénétrer dans le mystère des extensions non abéliennes; il n’est pas impossible que nous touchions là à des principes d’une fécondité extraordinaire, et que le premier pas décisif une fois fait dans cette voie doive nous ouvrir l’accès à de vastes domaines dont nous soupçonnons à peine l’existence; car jusqu’ici, pour amples que soient nos généralisations des résultats de Gauss, on ne peut dire que nous les ayons vraiment dépassés.”

Probably, the field with one element will appear in several talks with people such as Soule, Consani and Durov scheduled and topics such as Arakelov geometry (‘including the infinity-adic’) and ‘Generalized rings and schemes’ included.

Hopefully, more information will become available so that we can follow the workshop from a distant…

Little do our highly paid IC-experts know of the fact that MPG may also stand for the Max Planck Gesellschaft and that the string .mpg. is contained in all official email-addresses coming from the Max Planck Institute for Mathematics in Bonn…

This almost prematurely killed this blog.

It all started with Javier’s comment telling that they were planning to run a F_un seminar at the MPI. I had already promised to give a couple of talks at our ARTs seminar and as the Manin and Connes-Consani papers hit the arXiv a few days before, I thought it might be fun (no pun intended) to talk about them (or better, starting from them).

I asked Javier whether they were interested in setting up a blog that could serve as a vehicle to have some interaction between the two seminars. But I’ve made blog-proposals before leading nowhere, so I set up a small test-blog to show what I had in mind.

I copy/pasted a few older F_un posts from neverendingbooks to the new blog and didn’t care to remove links to other blogs which resulted in a few unwanted trackbacks in the comments of the Field with one element post over at the Secret Blogging Seminar.

Suddenly, the test-site got several hits daily coming from the SBS, while I wasn’t getting any feedback from the MPI, so after a week I chmodded the entry page and forgot the whole project.

Last friday afternoon, Javier used his Granada-account to ask whether I did receive his emails on the blog-project (quod non, via ‘mpg’). As their seminar was scheduled to start the next tuesday we decided on the spot to resurrect the test-blog and make it publicly available (we didn’t even have an ‘about’ page at the time…).

As math-blogging is a pretty small community, the F_un blog was inevitably discovered early on and thanks to generous links provided by David Corfield at the n-category cafe and Peter Woit at Not Even Wrong more people are coming here than we ever anticipated. Below the stats at 5 pm Brussels-time, precisely 1 week after start-up (you can see the effect of David’s evening-post on october 5th and Peter’s evening(in Europe)-post on october 8th).

Now that we’re pretty sure that everyone who might be interested in this blog knows of its existence, we ‘only’ have to provide content. Speaking of which, we are still looking for people willing to join the project!