Tag: Riemann

best of 2008 (2) : big theorems

Published January 3, 2009 by lieven

Charles Siegel of Rigorous Trivialities ran a great series on big theorems.

The series started january 10th 2008 with a post on Bezout’s theorem, followed by posts on Chow’s lemma, Serre duality, Riemann-Roch, Bertini, Nakayama’s lemma, Groebner bases, Hurwitz to end just before christmas with a post on Kontsevich’s formula.

Also at other blogs, 2008 was the year of series of long posts containing substantial pure mathematics.

Out of many, just two examples : Chris Schommer-Pries ran a three part series on TQFTs via planar algebras starting here, at the secret blogging seminar.
And, Peter Woit of Not Even Wrong has an ungoing series of posts called Notes on BRST, starting here. At the moment he is at episode nine.

It suffices to have a quick look at the length of any of these posts, to see that a great deal of work was put into these series (and numerous similar ones, elsewhere). Is this amount of time well spend? Or, should we focus on shorter, easier digestible math-posts?

What got me thinking was this merciless comment Charles got after a great series of posts leading up to Kontsevich’s formula :

“Perhaps you should make a New Years commitment to not be so obscurantist, like John Armstrong, and instead promote the public understanding of math!”

Well, if this doesn’t put you off blogging for a while, what will?

So, are we really writing the wrong sort of posts? Do math-blog readers only want short, flashy, easy reading posts these days? Or, is anyone out there taking notice of the hard work it takes to write such a technical post, let alone a series of them?

At first I was rather pessimistic about the probable answer to all these questions, but, fortunately we have Google Analytics to quantify things a bit.

Clearly I can only rely on the statistics for my own site, so I’ll treat the case of a recent post here : Mumford’s treasure map which tried to explain the notion of a generic point and how one might depict an affine scheme.

Here’s some of the Google Analytics data :

The yellow function gives the number of pageviews for that post, the value ranges between 0 and 600 (the number to the right of the picture). In total this post was viewed 2470 times, up till now.

The blue function tells the average time a visitor spend reading that post, the numbers range between 0 and 8 minutes (the times to the left of the picture). On average the time-on-page was 2.24 minutes, so in all people spend well over 92 hours reading this one post! This seems like a good return for the time it took me to write it…

Some other things can be learned from this data. Whereas the number of page-views has two peaks early on (one the day it was posted, the second one when Peter Woit linked to it) and is now steadily decreasing, the time-on-page for the later visitors is substantially longer than the early readers.

Some of this may be explained (see comment below) by returning visits. Here is a more detailed picture (orange = new visits, green=returning visits, blue=’total’ whatever this means).

All in all good news : there is indeed a market for longer technical math-posts and people (eventually) take time to read the post in detail.

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Andre Weil on the Riemann hypothesis

Published October 15, 2008 by lieven

Don’t be fooled by introductory remarks to the effect that ‘the field with one element was conceived by Jacques Tits half a century ago, etc. etc.’

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 $F_{1}$ . 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 $F_{1}$ ”. 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 $F_{1}$ . It is no accident that Connes makes a dramatic pauze in his YouTubeVideo to let the viewer see this equation on the backboard

$F_{1^{n}} \otimes_{F_{1}} Z = Z [x] / (x^{n} - 1)$

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 $F_{1}$ 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 $F_{1}$ . 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 $F_{1^{n}}$ .

Of course, $F_{1^{n}}$ does not exist in a rigorous sense, but we can think if a scheme $X$ contains n-th roots of unity, then it is defined over $F_{1^{n}}$ , so that there is a morphism

$p_{X} : X \to s p e c (F_{1^{n}}$

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 $l^{n}$ -th roots of unity, $l$ 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 $l$ ). 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 eu 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 gausienne 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’œil 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 nus les ayons vraiment dépassés.”

noncommutative F_un geometry (1)

Published October 13, 2008 by lieven

It is perhaps surprising that Alain Connes and Katia Consani, two icons of noncommutative geometry, restrict themselves to define commutative algebraic geometry over $F_{1}$ , the field with one element.

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 $A_{X}$ 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 $F_{1}$ -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

$N : abelian \to sets A \mapsto N (A)$

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to $N$ . 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 $F_{1}$ -geometry, which should start from a covariant functor

$N : groups \to sets$

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 $C$ -algebra $R$ together with a natural transformation (the ‘evaluation’)

$e : N \to maxi (R) N (G) \mapsto H o m_{C - a l g} (R, C G)$

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

The crucial extra ingredient is an affine $Z$ -algebra (possibly noncommutative) $S$
such that $N$ is a subfunctor of $mini (S) : G \mapsto H o m_{Z - a l g} (S, Z G)$ together with the following universal property :

any affine $Z$ -algebra $T$ having a gadget-morphism $(N, maxi (R)) \to (mini (T), maxi (T \otimes_{Z} C))$ comes from a $Z$ -algebra morphism $T \to S$ . (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 $groups \to sets$ still has as its integral form the integral torus $Z [x, x^{- 1}]$ . However, both theories quickly diverge beyond this example.

For example, consider the functor

$groups \to sets G \mapsto G \times G$

Then, if we restrict to abelian finite groups $abelian$ it is easy to see (again by a similar argument) that the two-dimensional integer torus $Z [x, y, x^{- 1}, y^{- 1}]$ 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 $Z$ -algebra map $ϕ : Z [x, y, x^{- 1}, y^{- 1}] \to Z G$ determines (and is determined by) a pair of commuting units in $Z G$ , so the above functor can not be a subfunctor if we allow non-Abelian groups.

But then, perhaps there isn’t a minimal integral $Z$ -form for this functor? Well, yes there is. Take the free group in two letters (that is, all words in noncommuting $x, y, x^{- 1}$ and $y^{- 1}$ satisfying only the trivial cancellation laws between a letter and its inverse), then the corresponding integral group-algebra $Z F_{2}$ does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map $f : T \mapsto C F_{2}$ and we have to show, using the universal property that the image of $T$ is contained in the integral group-algebra $Z F_{2}$ . Take a generator
$z$ of $T$ then the degree of the image $f (z)$ is bounded say by $d$ and we can always find a subgroup $H \subset F_{2}$ such that $F_{2} / H$ is a fnite group and the quotient map $C F_{2} \to C F_{2} / H$ is injective on the subspace spanned by all words of degree strictly less than $d + 1$ . Then, the usual diagram-chase finishes the proof.

What makes this work is that the free group $F_{2}$ has ‘enough’ subgroups of finite index, a property it shares with many interesting discrete groups. Whence the blurb-message :

if the integers $Z$ see a discrete group $Γ$ , then the field $F_{1}$ sees its profinite completion $\hat{Γ} = \underset{\leftarrow}{l i m} Γ / H$

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 $G a l (\overset{―}{Q} / Q)$ , 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 $\overset{―}{Q}$ -curve $X$ has a Belyi-map $X \to P_{\overset{―}{Q}}^{1}$ ramified only in three points $0, 1, \infty$ . The “drawing” corresponding to $X$ is a bipartite graph, drawn on the Riemann surface $X_{C}$ obtained by lifting the unit interval $[0, 1]$ to $X$ . 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 $G a l (\overset{―}{K} / K)$ , ou $K$ est une extension de type fini du corps premier) sur des groupes fondamentaux géométriques (profinis) de variétés algébriques (définies sur $K$ ), 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 ${\hat{π}}_{0, 3}$ , il y a le compactifié profini du groupe modulaire $S L_{2} (Z)$ , dont le quotient par le centre $\pm 1$ 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 $S L_{2} (Z)$ qui me fascine avant tout, est bien sur l’action extérieure du groupe de Galois $G a l (\overset{―}{Q} / Q)$ sur le compactifié profini. Par le théorème de Bielyi, prenant les compactifiés profinis de sous-groupes d’indice fini de $S L_{2} (Z)$ , et l’action extérieure induite (quitte à passer également à un sous-groupe overt de $G a l (\overset{―}{Q}, Q)$ ), on trouve essentiellement les groupes fondamentaux de toutes les courbes algébriques définis sur des corps de nombres $K$ , et l’action extérieure de $G a l (\overset{―}{K} / K)$ dessus.”

So, is there a noncommutative affine variety over $F_{1}$ of which the unique minimal integral model is the integral group algebra of the modular group $Z Γ$ (with $Γ = P S L_{2} (Z)$ ? Yes, here it is

$N_{Γ} : groups \to sets G \mapsto G_{2} \times G_{3}$

where $G_{n}$ is the set of all elements of order $n$ in $G$ . The reason behind this is that the modular group is the free group product $C_{2} * C_{3}$ .

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 $F_{1}$ -scheme are exactly the (modular) dessins d’enfants…