In his 1951 report [Sur la théorie du corps de classes](https://projecteuclid.org/euclid.jmsj/1261734944), 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](https://mathoverflow.net/questions/41296/lun-des-probl%C3%A8mes-fondamentaux-de-la-th%C3%A9orie-des-nombres#comment97379_41296), a translation (by [@TonyScholl](https://mathoverflow.net/questions/41296/lun-des-probl%C3%A8mes-fondamentaux-de-la-th%C3%A9orie-des-nombres#comment97392_41296)):



> 
> 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?



L'un des problèmes fondamentaux de la théorie des nombres
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.


