Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques

The main purpose of this dissertation is to introduce Shimura curves from the non-Archimedean point of view, paying special attention to those aspects that can make this theory amenable for computations. Despite the fact that the theory of p-adic uniformization of Shimura curves goes back to the 196...

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Autor: Milione, Piermarco
Tipo de recurso: tesis doctoral
Estado:Versión publicada
Fecha de publicación:2016
País:España
Institución:CBUC, CESCA
Repositorio:TDR. Tesis Doctorales en Red
OAI Identifier:oai:www.tdx.cat:10803/402209
Acceso en línea:http://hdl.handle.net/10803/402209
Access Level:acceso abierto
Palabra clave:Corbes algebraiques
Curvas algebraicas
Algebraic curves
Geometria algebraica
Geometría algebraica
Algebraic geometry
Ciències Experimentals i Matemàtiques
512
514
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spelling Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiquesMilione, PiermarcoCorbes algebraiquesCurvas algebraicasAlgebraic curvesGeometria algebraicaGeometría algebraicaAlgebraic geometryCiències Experimentals i Matemàtiques512514The main purpose of this dissertation is to introduce Shimura curves from the non-Archimedean point of view, paying special attention to those aspects that can make this theory amenable for computations. Despite the fact that the theory of p-adic uniformization of Shimura curves goes back to the 1960s with the results of Cerednik and Drinfeld, only in the last years explicit examples related to these uniformizations have been computed. The structure of this dissertation is as follows. In Chapter 1 we introduce Shimura curves starting from an indefinite quaternion algebra H over a totally real field F. This is done mostly following the fundamental paper of Shimura [Shi67]. We also give the definitions using the adelic approach of [Shi70b] and [Shi70c]. The point of view we adopt is the arithmetical one, since we try to make clear the link connecting Shimura curves to the arithmetic of quaternion algebras. In this sense, we give evidence of why Shimura curves have to be considered a geometric interpretation of most arithmetical phenomena in quaternion orders. Chapter 2 has the aim of introducing those non-Archimedean objects which appear later in the statements of the theorems of Cerednik and Drinfeld. In Chapter 3 we start the study of fundamental domains in Hp for the action of discrete and cocompact subgroups of PGL2(Qp) arising in the p-adic uniformization of Shimura curves. In Chapter 4 we associate to the p-adic uniformization of the Shimura curve X(DH;N) certain parameters in Hp(Cp) analogous to the complex multiplication parameters in H: we refer to them by p-imaginary multiplication paramters, since they are defined over the unramified quadratic extension of Qp. In the study of these parameters, we follow the p-adic analog of the line adopted in [AB04]. Specifically, we are able to recover these parameters as zeros of certain binary quadratic forms with p-adic coefficients.Universitat de BarcelonaBayer i Isant, PilarUniversitat de Barcelona. Departament d'Àlgebra i Geometria201720172016info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersion191 p.application/pdfapplication/pdfhttp://hdl.handle.net/10803/402209TDX (Tesis Doctorals en Xarxa)reponame:TDR. Tesis Doctorales en Redinstname:CBUC, CESCAInglésL'accés als continguts d'aquesta tesi queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:www.tdx.cat:10803/4022092026-06-14T12:46:07Z
dc.title.none.fl_str_mv Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
title Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
spellingShingle Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
Milione, Piermarco
Corbes algebraiques
Curvas algebraicas
Algebraic curves
Geometria algebraica
Geometría algebraica
Algebraic geometry
Ciències Experimentals i Matemàtiques
512
514
title_short Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
title_full Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
title_fullStr Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
title_full_unstemmed Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
title_sort Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
dc.creator.none.fl_str_mv Milione, Piermarco
author Milione, Piermarco
author_facet Milione, Piermarco
author_role author
dc.contributor.none.fl_str_mv Bayer i Isant, Pilar
Universitat de Barcelona. Departament d'Àlgebra i Geometria
dc.subject.none.fl_str_mv Corbes algebraiques
Curvas algebraicas
Algebraic curves
Geometria algebraica
Geometría algebraica
Algebraic geometry
Ciències Experimentals i Matemàtiques
512
514
topic Corbes algebraiques
Curvas algebraicas
Algebraic curves
Geometria algebraica
Geometría algebraica
Algebraic geometry
Ciències Experimentals i Matemàtiques
512
514
description The main purpose of this dissertation is to introduce Shimura curves from the non-Archimedean point of view, paying special attention to those aspects that can make this theory amenable for computations. Despite the fact that the theory of p-adic uniformization of Shimura curves goes back to the 1960s with the results of Cerednik and Drinfeld, only in the last years explicit examples related to these uniformizations have been computed. The structure of this dissertation is as follows. In Chapter 1 we introduce Shimura curves starting from an indefinite quaternion algebra H over a totally real field F. This is done mostly following the fundamental paper of Shimura [Shi67]. We also give the definitions using the adelic approach of [Shi70b] and [Shi70c]. The point of view we adopt is the arithmetical one, since we try to make clear the link connecting Shimura curves to the arithmetic of quaternion algebras. In this sense, we give evidence of why Shimura curves have to be considered a geometric interpretation of most arithmetical phenomena in quaternion orders. Chapter 2 has the aim of introducing those non-Archimedean objects which appear later in the statements of the theorems of Cerednik and Drinfeld. In Chapter 3 we start the study of fundamental domains in Hp for the action of discrete and cocompact subgroups of PGL2(Qp) arising in the p-adic uniformization of Shimura curves. In Chapter 4 we associate to the p-adic uniformization of the Shimura curve X(DH;N) certain parameters in Hp(Cp) analogous to the complex multiplication parameters in H: we refer to them by p-imaginary multiplication paramters, since they are defined over the unramified quadratic extension of Qp. In the study of these parameters, we follow the p-adic analog of the line adopted in [AB04]. Specifically, we are able to recover these parameters as zeros of certain binary quadratic forms with p-adic coefficients.
publishDate 2016
dc.date.none.fl_str_mv 2016
2017
2017
dc.type.none.fl_str_mv info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/publishedVersion
format doctoralThesis
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10803/402209
url http://hdl.handle.net/10803/402209
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 191 p.
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Universitat de Barcelona
publisher.none.fl_str_mv Universitat de Barcelona
dc.source.none.fl_str_mv TDX (Tesis Doctorals en Xarxa)
reponame:TDR. Tesis Doctorales en Red
instname:CBUC, CESCA
instname_str CBUC, CESCA
reponame_str TDR. Tesis Doctorales en Red
collection TDR. Tesis Doctorales en Red
repository.name.fl_str_mv
repository.mail.fl_str_mv
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