TITLE: L'un des problèmes fondamentaux de la théorie des nombres
QUESTION [17 upvotes]: 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?

REPLY [3 votes]: 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.