Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positiv...

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Detalles Bibliográficos
Autores: Girondo Sirvent, Ernesto, González Díez, Gabino, González Jiménez, Enrique, Steuding, R., Steuding, J.
Tipo de recurso: artículo
Fecha de publicación:2009
País:España
Institución:Universidad Autónoma de Madrid
Repositorio:Biblos-e Archivo. Repositorio Institucional de la UAM
Idioma:inglés
OAI Identifier:oai:repositorio.uam.es:10486/711109
Acceso en línea:http://hdl.handle.net/10486/711109
https://dx.doi.org/10.2478/s12175-009-0126-3
Access Level:acceso abierto
Palabra clave:Congruent Number Problem
Elliptic Curves
Number Fields
Matemáticas
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spelling Right triangles with algebraic sides and elliptic curves over number fieldsGirondo Sirvent, ErnestoGonzález Díez, GabinoGonzález Jiménez, EnriqueSteuding, R.Steuding, J.Congruent Number ProblemElliptic CurvesNumber FieldsMatemáticasGiven any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 - n 2 X over ℚ(λ). © 2009 © Versita Warsaw and Springer-Verlag Berlin HeidelbergResearch of the third author was supported in part by grant MTM 2006-10548 (Ministerio de Educación y Ciencia, Spain) and CCG06-UAM/ESP-0477 (Universidad Autónoma de Madrid – Comunidad de Madrid, Spain). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain)De GruyterDepartamento de MatemáticasFacultad de Ciencias20092009-05-17research articlehttp://purl.org/coar/resource_type/c_2df8fbb1VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10486/711109https://dx.doi.org/10.2478/s12175-009-0126-3reponame:Biblos-e Archivo. Repositorio Institucional de la UAMinstname:Universidad Autónoma de MadridInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:repositorio.uam.es:10486/7111092026-06-23T12:46:27Z
dc.title.none.fl_str_mv Right triangles with algebraic sides and elliptic curves over number fields
title Right triangles with algebraic sides and elliptic curves over number fields
spellingShingle Right triangles with algebraic sides and elliptic curves over number fields
Girondo Sirvent, Ernesto
Congruent Number Problem
Elliptic Curves
Number Fields
Matemáticas
title_short Right triangles with algebraic sides and elliptic curves over number fields
title_full Right triangles with algebraic sides and elliptic curves over number fields
title_fullStr Right triangles with algebraic sides and elliptic curves over number fields
title_full_unstemmed Right triangles with algebraic sides and elliptic curves over number fields
title_sort Right triangles with algebraic sides and elliptic curves over number fields
dc.creator.none.fl_str_mv Girondo Sirvent, Ernesto
González Díez, Gabino
González Jiménez, Enrique
Steuding, R.
Steuding, J.
author Girondo Sirvent, Ernesto
author_facet Girondo Sirvent, Ernesto
González Díez, Gabino
González Jiménez, Enrique
Steuding, R.
Steuding, J.
author_role author
author2 González Díez, Gabino
González Jiménez, Enrique
Steuding, R.
Steuding, J.
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Departamento de Matemáticas
Facultad de Ciencias
dc.subject.none.fl_str_mv Congruent Number Problem
Elliptic Curves
Number Fields
Matemáticas
topic Congruent Number Problem
Elliptic Curves
Number Fields
Matemáticas
description Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 - n 2 X over ℚ(λ). © 2009 © Versita Warsaw and Springer-Verlag Berlin Heidelberg
publishDate 2009
dc.date.none.fl_str_mv 2009
2009-05-17
dc.type.none.fl_str_mv research article
http://purl.org/coar/resource_type/c_2df8fbb1
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv http://hdl.handle.net/10486/711109
https://dx.doi.org/10.2478/s12175-009-0126-3
url http://hdl.handle.net/10486/711109
https://dx.doi.org/10.2478/s12175-009-0126-3
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv De Gruyter
publisher.none.fl_str_mv De Gruyter
dc.source.none.fl_str_mv reponame:Biblos-e Archivo. Repositorio Institucional de la UAM
instname:Universidad Autónoma de Madrid
instname_str Universidad Autónoma de Madrid
reponame_str Biblos-e Archivo. Repositorio Institucional de la UAM
collection Biblos-e Archivo. Repositorio Institucional de la UAM
repository.name.fl_str_mv
repository.mail.fl_str_mv
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