Andre Weil on the Riemann hypothesis

Posted by on Oct 12, 2008 in featured, outreach7 comments

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 \mathbb{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 \mathbb{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 \mathbb{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

\mathbb{F}_{1^n} \otimes_{\mathbb{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 \mathbb{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 \mathbb{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 \mathbb{F}_{1^n}.

Of course, \mathbb{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 \mathbb{F}_{1^n}, so that there is a morphism

p_X~:~X \rightarrow spec(\mathbb{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 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.”

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  1. Hah!! I’m no mathematician, but that was pretty obvious! Hmmm, I see. So I’m supposed to shut up and not mention the Riemann hypothesis? That’s a bit tricky with zeta values lying all over the cutting room floor and particle masses lining up neatly in nice patterns.

  2. ????? Ive reread the post just now and i cannot understand where you got that ” So I’m supposed to shut up and not mention the Riemann hypothesis?” from.
    Perhaps you took the bold sentence on F_un mathematics as a mission statement of this blog (I’d wish it were…) but then i didnt express myself well. When i say ‘F_un mathematics’ in such a context i mean the collective work of all people interested in the field with one element. Perhaps we should change the title of this blog to CNPUC (ceci n’est pas un corps) anyway, as some people already do.
    So, please go on writing exciting posts teaching ‘us mathematicians’ the connections between the RH and physics!

  3. It was the statement: few of the authors will be willing to let you in on the secret… and the implied professional constraints. But please keep the blog title as is … I’m already tired of people pointing out the reference to the pipe.

  4. Lol, my own motivations are waaaaay less elevated than RH, and can be summarized as this: enter an emerging field, publish n papers before it gets too crowded, so that I can obtain my next post-doc with some guarantees, and then take my rucksack and walk to the next village as soon as there are too many people around. But also, I think I am the only one who didn’t understand the reference to the pipe until someone pointed it out… shame on my artistic culture!

  5. gausienne –> gaussienne, nus –> nous

  6. après qu’on eu perdu –> après qu’on eut perdu.

    Let no one add a circumflex on the “u” of “eut”: “après que” is followed by the indicative, not the conjunctive. The conjunctive IS often used nowadays, though. Probably because of a false analogy with “avant que”, which is indeed followed by the conjunctive. The classical rule follows the flawless logic that the past is certain but not the future (maybe medieval grammarians knew quantum mechanics ?)

    Happy New Year to you, Lieven, and to all those who have the good taste to read your blog.

  7. @chandan, @georges : it took me a bit to figure out what’s going on, but i discovered the MathOverflow topic and answer now… and read through the comments

    I thank Ed Dean for linking to the Fun-post, Chandan for correcting the misspellings and Georges for the kind words. I agree with Georges that a cut&copy of a blogpost-quoted text does not require a link to that post (though it is always much appreciated). It is rewarding to see these old posts getting a second chance…

    I wish you all a lot of mathematical (and other) fun in 2011 :: lieven.

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