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Andre Weil on the Riemann hypothesis

Posted By lievenlb On October 12, 2008 @ 9:51 pm In featured,outreach | 7 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 . 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 [1].

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 [2] of most authors is to get a grip on cyclotomy over . It is no accident that Connes makes a dramatic pauze in his YouTubeVideo [3] 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 [4]. 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 :

Les automorphismes induits sur les groupes de classes par les automorphismes du corps, les propriÃ©tÃ©s des restes de normes dans les cas non cycliques, le passage Ã  la limite (inductive ou projective) quand on remplace le corps de base par des extensions, par example cyclotomiques, de degrÃ© indÃ©finiment croissant, sont autant de questions sur lesquelles notre ignorance est Ã  peu prÃ¨s complÃ¨te, et dont l’Ã©tude contient peut-Ãªtre la clef de l’hypothese de Riemann; Ã©troitement liÃ©e Ã  celles-ci est l’Ã©tude du conducteur d’Artin, et en particulier, dans le cas local, la recherche de la reprÃ©sentation dont la trace s’exprime au moyen des caractÃ¨res simples avec des coefficients Ã©gaux aux exposants de leurs conducteurs.

URL to article: http://cage.ugent.be/~kthas/Fun/index.php/andre-weil-on-the-rh.html

URLs in this post:

[1] Riemann Hypothesis: http://en.wikipedia.org/wiki/Riemann_hypothesis

[2] Jugendtraum: http://planetmath.org/encyclopedia/KroneckersJugendtraum.html