Characterization of subfields of adelic algebras by a product formula

Abstract

[EN]We consider projective, irreducible, non-singular curves over an algebraically closed field k. A cover Y → X of such curves corresponds to an extension / of their function fields and yields an isomorphism AY ≃ AX ⊗ of their geometric adele rings. The primitive element theorem shows that AY is a quotient of AX[T] by a polynomial. In general, we may look at quotient algebras AX {p} = AX [T ]/(p(T )) where p(T ) ∈ AX [T ] is monic and separable over AX , and try to characterize the field extensions / lying in AX {p} which arise from covers as above. We achieve this in two ways; the first, topologically, as those which embed discretely in AX {p}. The second is the characterization of such subfields 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 AX {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.