I am a total non-expert, so the answer to this question may be obvious, but here goes.

In Chevalley's formulation of CFT we get Artin maps $J_k \rightarrow Gal(L/k)$, where $J_k$ is the group of all ideles of $k$. However, we know there is a nice subgroup $J_k^1$ of the ideles obtained by taking only those satisfying the product formula $\prod_{v} |x_v| = 1$. Note that this still contains all the principal ideles, still surjects onto $I_k$ and has additional attractive properties like the compactness of $J_k^1/k^*$. Is there a way to set up CFT using quotients of this nicer group, and if so, what are the advantages of working with the superficially more unwieldy $J_k$?

Thanks.