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) :-)