Characterization of subfields of adelic algebras by a product formula

Abstract We consider projective, irreducible, non-singular curves over an algebraically closed field $$\Bbbk $$ k . A cover $$Y \rightarrow X$$ Y → X of such curves corresponds to an extension $$\Omega /\Sigma $$ Ω / Σ of their function fields and yields an isomorphism $$\mathbb {A}_{Y} \simeq \mathbb {A}_{X} \otimes _{\Sigma } \Omega $$ A Y ≃ A X ⊗ Σ Ω of their geometric adele rings. The primitive element theorem shows that $$\mathbb {A}_{Y}$$ A Y is a quotient of $$\mathbb {A}_{X}[T]$$ A X [ T ] by a polynomial. In general, we may look at quotient algebras $$\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace = \mathbb {A}_{X}[T]/(\mathbbm {p}(T))$$ A X { p } = A X [ T ] / ( p ( T ) ) where $$\mathbbm {p}(T) \in \mathbb {A}_{X}[T]$$ p ( T ) ∈ A X [ T ] is monic and separable over $$\mathbb {A}_{X}$$ A X , and try to characterize the field extensions $$\Omega /\Sigma $$ Ω / Σ lying in $$\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace $$ A X { p } which arise from covers as above. We achieve this in two ways; the first, topologically, as those $$\Omega $$ Ω which embed discretely in $$\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace $$ A X { p } . The second is the characterization of such subfields $$\Omega $$ Ω as those which satisfy the additive analog of the product formula in classical adele rings. The technical machinery is based on the use of Tate topologies on the quotient algebras $$\mathbb {A}_{X}\lbrace \hspace{0.0pt}{\mathbbm {p}}\hspace{0.0pt}\rbrace $$ A X { p } . These are not locally compact, but we are able to define an additive content function as an index measuring the discrepancy of dimensions in commensurable subspaces.