Characterization of subfields of adelic algebras by a product formula

We consider projective, irreducible, non-singular curves over an algebraically closed field $\k$. A cover $Y \to X$ of such curves corresponds to an extension $\Omega/\Sigma$ of their function fields and yields an isomorphism $\A_{Y} \simeq \A_{X} \otimes_{\Sigma} \Omega$ of their geometric adele rings. The primitive element theorem shows that $\A_{Y}$ is a quotient of $\A_{X}[T]$ by a polynomial. In general, we may look at quotient algebras $\AXp{\p} = \A_{X}[T]/(\p(T))$ where $\p(T) \in \A_{X}[T]$ is monic and separable over $\A_{X}$, and try to characterize the field extensions $\Omega/\Sigma$ lying in $\AXp{\p}$ which arise from covers as above. We achieve this topologically, namely, as those $\Omega$ which embed discretely in $\AXp{\p}$, and in terms of an additive analog of the product formula for global fields, a result which is reminiscent of classical work of Artin-Whaples and Iwasawa. The technical machinery requires studying which topology on $\AXp{\p}$ is natural for this problem. Local compactness no longer holds, but instead we have linear topologies defined by commensurability of $\k$-subspaces which coincide with the restricted direct product topology with respect to integral closures. The content function is given as an index measuring the discrepancy in commensurable subspaces.