Connes reformulates Weil's question <a href="http://www.alainconnes.org/docs/ramis.pdf" title="pdf">here</a> 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 <a href="http://www.alainconnes.org/docs/imufinal.pdf" title="pdf">here</a>. 

Edit: Morava wrote on <a href="http://www.springerlink.com/content/k69721u35680h843/" title="link">Weil group representations coming from algebraic topology</a>. 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) :-)