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.