Ribet Bimodules and the Specialization of Heegner points

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X0(D,N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P in CM(R) specializes to a singular point (resp. a connected componen...

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Detalhes bibliográficos
Autor: Molina Blanco, Santiago|||0000-0001-9420-2807
Tipo de documento: artigo
Data de publicação:2012
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositório:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglês
OAI Identifier:oai:upcommons.upc.edu:2117/86382
Acesso em linha:https://hdl.handle.net/2117/86382
Access Level:Acceso aberto
Palavra-chave:Algebra
Shimura varieties
Curves, Elliptic
Arithmetical
Aritmètica
Varietats de Shimura
Corbes modulars
Àlgebra
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
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oai_identifier_str oai:upcommons.upc.edu:2117/86382
network_acronym_str ES
network_name_str España
repository_id_str
spelling Ribet Bimodules and the Specialization of Heegner pointsMolina Blanco, Santiago|||0000-0001-9420-2807AlgebraShimura varietiesCurves, EllipticArithmeticalAritmèticaVarietats de ShimuraCorbes modularsÀlgebraÀrees temàtiques de la UPC::Matemàtiques i estadística::ÀlgebraFor a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X0(D,N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P in CM(R) specializes to a singular point (resp. a connected component) of the special fiber Xp of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X0(D,N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebras that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in Xp. We also illustrate our results with an explicit example.20122012-01-0120162016-04-28journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/86382reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/863822026-05-27T15:37:01Z
dc.title.none.fl_str_mv Ribet Bimodules and the Specialization of Heegner points
title Ribet Bimodules and the Specialization of Heegner points
spellingShingle Ribet Bimodules and the Specialization of Heegner points
Molina Blanco, Santiago|||0000-0001-9420-2807
Algebra
Shimura varieties
Curves, Elliptic
Arithmetical
Aritmètica
Varietats de Shimura
Corbes modulars
Àlgebra
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
title_short Ribet Bimodules and the Specialization of Heegner points
title_full Ribet Bimodules and the Specialization of Heegner points
title_fullStr Ribet Bimodules and the Specialization of Heegner points
title_full_unstemmed Ribet Bimodules and the Specialization of Heegner points
title_sort Ribet Bimodules and the Specialization of Heegner points
dc.creator.none.fl_str_mv Molina Blanco, Santiago|||0000-0001-9420-2807
author Molina Blanco, Santiago|||0000-0001-9420-2807
author_facet Molina Blanco, Santiago|||0000-0001-9420-2807
author_role author
dc.subject.none.fl_str_mv Algebra
Shimura varieties
Curves, Elliptic
Arithmetical
Aritmètica
Varietats de Shimura
Corbes modulars
Àlgebra
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
topic Algebra
Shimura varieties
Curves, Elliptic
Arithmetical
Aritmètica
Varietats de Shimura
Corbes modulars
Àlgebra
Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra
description For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X0(D,N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P in CM(R) specializes to a singular point (resp. a connected component) of the special fiber Xp of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X0(D,N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebras that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in Xp. We also illustrate our results with an explicit example.
publishDate 2012
dc.date.none.fl_str_mv 2012
2012-01-01
2016
2016-04-28
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
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 https://hdl.handle.net/2117/86382
url https://hdl.handle.net/2117/86382
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

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
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

http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300724

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