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?