Characterization of subfields of adelic algebras by a product formula

[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...

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Detalles Bibliográficos
Autores: Navas Vicente, Luis Manuel, Plaza Martín, Francisco José
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2025
País:España
Institución:Universidad de Salamanca (USAL)
Repositorio:GREDOS. Repositorio Institucional de la Universidad de Salamanca
OAI Identifier:oai:gredos.usal.es:10366/167360
Acceso en línea:http://hdl.handle.net/10366/167360
Access Level:acceso abierto
Palabra clave:Algebraic curves
Characterization of function fields
Algebras over adele ring
Product formula
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spelling Characterization of subfields of adelic algebras by a product formulaNavas Vicente, Luis ManuelPlaza Martín, Francisco JoséAlgebraic curvesCharacterization of function fieldsAlgebras over adele ringProduct formula[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.PID2023-150787NB-I00Springer202520252025info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://hdl.handle.net/10366/167360reponame:GREDOS. Repositorio Institucional de la Universidad de Salamancainstname:Universidad de Salamanca (USAL)InglésPID2023-150787NB-I00info:eu-repo/semantics/openAccessoai:gredos.usal.es:10366/1673602026-06-07T06:28:51Z
dc.title.none.fl_str_mv Characterization of subfields of adelic algebras by a product formula
title Characterization of subfields of adelic algebras by a product formula
spellingShingle Characterization of subfields of adelic algebras by a product formula
Navas Vicente, Luis Manuel
Algebraic curves
Characterization of function fields
Algebras over adele ring
Product formula
title_short Characterization of subfields of adelic algebras by a product formula
title_full Characterization of subfields of adelic algebras by a product formula
title_fullStr Characterization of subfields of adelic algebras by a product formula
title_full_unstemmed Characterization of subfields of adelic algebras by a product formula
title_sort Characterization of subfields of adelic algebras by a product formula
dc.creator.none.fl_str_mv Navas Vicente, Luis Manuel
Plaza Martín, Francisco José
author Navas Vicente, Luis Manuel
author_facet Navas Vicente, Luis Manuel
Plaza Martín, Francisco José
author_role author
author2 Plaza Martín, Francisco José
author2_role author
dc.subject.none.fl_str_mv Algebraic curves
Characterization of function fields
Algebras over adele ring
Product formula
topic Algebraic curves
Characterization of function fields
Algebras over adele ring
Product formula
description [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.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10366/167360
url http://hdl.handle.net/10366/167360
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv PID2023-150787NB-I00
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:GREDOS. Repositorio Institucional de la Universidad de Salamanca
instname:Universidad de Salamanca (USAL)
instname_str Universidad de Salamanca (USAL)
reponame_str GREDOS. Repositorio Institucional de la Universidad de Salamanca
collection GREDOS. Repositorio Institucional de la Universidad de Salamanca
repository.name.fl_str_mv
repository.mail.fl_str_mv
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