L'un des problèmes fondamentaux de la théorie des nombres

Question

In his 1951 report Sur la théorie du corps de classes, Weil writes that

La recherche d'une interprétation de $C_k$ si $k$ est un corps de nombres, analogue en quelque manière à l'interprétation par un groupe de Galois quand $k$ est un corps de fonctions, me semble constituer l'un des problèmes fondamentaux de la théorie des nombres à l'heure actuelle; il se peut qu'une telle interprétation renferme la clef de l'hypothèse de Riemann ….

As requested by @PeteL.Clark, a translation (by @TonyScholl):

The search for an interpretation for $C_k$, where $k$ is a number field—in some way analogous to its interpretation by a Galois group when $k$ is a function field—seems to me to be one of the fundamental problems of number theory today; perhaps such an interpretation contains the key to the Riemann hypothesis ….

Here, $C_k$ is of course the idèle class group of the number field $k$.

I've heard that some people working in noncommutative geometry have thought about this problem.

Question. What progress has since been made towards such an interpretation?

Answer 1

Interpreting "progress" in a different (perhaps more controversial!) way than in Tony Scholl's answer, one could also mention that Langlands (followed by Kottwitz and perhaps others) has introduced a hypothetical group $L_k$ which should be even bigger than $W_k$, normally referred to as the Langlands group, which should bear the same relationship to arbitrary automorphic forms as $C_k$ does to Grossencharacters.

It might help to remark that the algebraic Grossencharacters correspond geometrically to abelian varieties with CM, and so the algebraic envelope of $C_k$ (the associated pro-algebraic torus through which all algebraic Grossencharacters factor) is the Tannakian group of the category of motives over $k$ generated by the Tate motive together with the motives of all abelian varieties over $k$ which have CM defined over $k$.

The algebraic envelope of $W_k$ (which is now a non-commutative reductive pro-algebraic group) is the Tannakian group of the category of motives over $k$ generated by motives which are potentially CM, i.e. which become CM motives (i.e. belong to the category considered in the preceding paragraph, i.e. are classified by an algebraic Grossencharacter on $C_l$) over some extension $l$ of $k$. This category contains all Artin motives, for example.

The algebraic envelope of $L_k$ should be the Tannakian group of the category of all motives over $k$.

So the problem of constructing $L_k$ can be thought of, from this point of view, as the problem of enlarging the category of motives so that one can make sense of motives with "non-integral Hodge grading" (i.e. has $h^{p,q}$ for $p$ and $q$ complex numbers that are not necessarily integral); $L_k$ would then be (some version of) the Tannakian group of this category.

Going back to $C_k$, from this optic one would like to generalize the notion of CM abelian variety to include objects with non-integral Hodge gradings, which would give rise to non-algebraic Grossencharacters in the way that usual CM abelian varieties correspond to algebraic Grossencharacters.

And of course, for $W_k$, once wants to generalize potentially CM abelian varieties in the same way.

Answer 2

Although this isn't at the moment heading towards the Riemann hypothesis, the most promising line of research in this area seems to be Lichtenbaum's Weil-etale cohomology. In the 1951 paper you cite, Weil constructed, for any finite Galois extension $K/k$, a topological group $W_{K/k}$ which is an extension of the Galois group $\rm{Gal}(K^{ab}/k)$ by the connected component $D_K$ of $C_K$. (See sec.11.6 of Tate's article in Cassels-Frohlich, or his article in vol.2 of Corvallis). This inverse limit of the groups $W_{K/k}$ over all $K/k$ is the Weil group $W_k$ of $k$, a huge topological group.

Lichtenbaum's idea is to interpret special values of $L$-functions using a suitable cohomology theory for $W_k$. In fact one wants to work with some sort of sheaves over the ring of integers of $k$, not just over $\mathop{\rm{Spec}}(k)$. In the function field case, there is a well-developed theory, initiated by Lichtenbaum and worked out in considerable generality by Geisser. (Function fields are easier because then $D_k$ is trivial). For number fields, things are much harder but there has been recent progress by Baptiste Morin - see his 2 recent preprints on arxiv.

Answer 3

Connes reformulates Weil's question here as: "Is there a non trivial Brauer theory of central simple algebras over $\mathbb{C}$ ?" and tells his solution of his reformulation in the context of a "cosmic galois group" in renormalization here.

Edit: Morava wrote on Weil group representations coming from algebraic topology. The bibl. list at the end of his article shows Weil's article with the question on $C_k$. It would be great if someone would look at it and tell more about it , please understandable for a non-(algebraic topologist) :-)

Edit: Lieven le Bruyn runs a seminar on "a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program", and will post lecture notes on his blog if enough people are interested.