Rational points on Shimura curves and Galois representations
This thesis explores one of the essential arithmetical and diophantine properties of Shimura curves and their Atkin-Lehner quotients: the existence of rational points on these families of curves over both number fields and their completions. Due to their moduli interpretation, Shimura curves (and mo...
| Autor: | |
|---|---|
| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/284400 |
| Acceso en línea: | http://hdl.handle.net/10803/284400 https://dx.doi.org/10.5821/dissertation-2117-95533 |
| Access Level: | acceso abierto |
| Palabra clave: | Shimura curves Atkin-Lehner quotients Rational points Hasse principle Brauer-Manin obstruction Galois representations 51 |
| id |
ES_5f2deb34be718e4cb890ae43ecd31abe |
|---|---|
| oai_identifier_str |
oai:www.tdx.cat:10803/284400 |
| network_acronym_str |
ES |
| network_name_str |
España |
| repository_id_str |
|
| dc.title.none.fl_str_mv |
Rational points on Shimura curves and Galois representations |
| title |
Rational points on Shimura curves and Galois representations |
| spellingShingle |
Rational points on Shimura curves and Galois representations Vera Piquero, Carlos de Shimura curves Atkin-Lehner quotients Rational points Hasse principle Brauer-Manin obstruction Galois representations 51 |
| title_short |
Rational points on Shimura curves and Galois representations |
| title_full |
Rational points on Shimura curves and Galois representations |
| title_fullStr |
Rational points on Shimura curves and Galois representations |
| title_full_unstemmed |
Rational points on Shimura curves and Galois representations |
| title_sort |
Rational points on Shimura curves and Galois representations |
| dc.creator.none.fl_str_mv |
Vera Piquero, Carlos de |
| author |
Vera Piquero, Carlos de |
| author_facet |
Vera Piquero, Carlos de |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Rotger, Víctor Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II |
| dc.subject.none.fl_str_mv |
Shimura curves Atkin-Lehner quotients Rational points Hasse principle Brauer-Manin obstruction Galois representations 51 |
| topic |
Shimura curves Atkin-Lehner quotients Rational points Hasse principle Brauer-Manin obstruction Galois representations 51 |
| description |
This thesis explores one of the essential arithmetical and diophantine properties of Shimura curves and their Atkin-Lehner quotients: the existence of rational points on these families of curves over both number fields and their completions. Due to their moduli interpretation, Shimura curves (and modular curves) are of great arithmetic significance. The research line started by the work of Mazur on rational points on modular curves, leading to the classification of rational torsion subgroups of elliptic curves over Q, has been intensively and successfuly explored by many authors, and the general philosophy is that rational points on modular and Shimura curves over number fields should correspond only to CM-points, except for a few exceptional cases. Aiming to provide more evidence in support of this philosophy, in this thesis we propose new approaches for studying the lack of rational points over number fields on Shimura curves and their Atkin-Lehner quotients. Furthermore, we also wish to show that these curves provide a wealth of counterexamples to the Hasse principle, hence they can be used to test cohomological obstructions to this local-global principle, as for example the Brauer-Manin obstruction. The thesis is divided into two parts. The first of them is devoted to study the arithmetic and the geometry of the cyclic Galois coverings of Shimura curves introduced by Jordan. On the one hand, we determine the group of modular automorphisms of the Shimura curves arising from these coverings, showing in particular that Atkin-Lehner involutions can be lifted through them. As a consequence, we can produce cyclic étale coverings of Atkin-Lehner quotients of Shimura curves, which can be used to study the (non-)existence of rational points on these curves by applying descent techniques. Further, we characterise the existence of local points at bad reduction primes on both the intermediate curves of Jordan's coverings and their quotients by Atkin-Lehner involutions. This part of the thesis exploits the adèlic formalism of Shimura curves, as well as the padic uniformisation theory of Cerednik and Drinfeld, generalising previous work of Jordan-Livné and Ogg. In the second part of the thesis, we propose and investigate a method for proving the non-existence of rational points over a number field K on a coarse moduli space X of abelian varieties with additional structure, with special interest in cases where the moduli problem is not fine and K-rational points may not be represented by abelian varieties admitting a model over K (which is the generic situation if the abelian varieties being classified have even dimension). The original inspiration dates back to the works of Mazur and Jordan, in which the authors study the existence K-rational points on modular and Shimura curves, respectively, through the Galois representations attached to the elliptic curves and abelian surfaces parametrised by them. However, they need to assume these varieties to be defined over K (their field of moduli), a hypothesis which need not be correlated to the non-existence of K-rational points on the moduli space. To overcome this, we attach Galois representations to K-rational points on X rather than to the abelian varieties classified by them (what we call "Galois representations over fields of moduli"), inspired by the work of Ellenberg and Skinner on the modularity of Q-curves. We exemplify our method, combined with the cyclic coverings studied in the first part of the thesis, in the case where X is either a Shimura curve or an Atkin-Lehner quotient of it and K is an imaginary quadratic field or the field of rational numbers, respectively. In both cases, we produce new counterexamples to the Hasse principle. And moreover, in the first case we prove that these counterexamples are accounted for by the Brauer-Manin obstruction. The results of this second part complement previous work of Parent-Yafaev, Gillibert or Clark, for example. |
| publishDate |
2014 |
| dc.date.none.fl_str_mv |
2014 2014 2014 |
| 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/10803/284400 https://dx.doi.org/10.5821/dissertation-2117-95533 |
| url |
http://hdl.handle.net/10803/284400 https://dx.doi.org/10.5821/dissertation-2117-95533 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.rights.none.fl_str_mv |
http://creativecommons.org/licenses/by-nc-sa/3.0/es/ info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
http://creativecommons.org/licenses/by-nc-sa/3.0/es/ |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
144 p. application/pdf application/pdf |
| dc.publisher.none.fl_str_mv |
Universitat Politècnica de Catalunya |
| publisher.none.fl_str_mv |
Universitat Politècnica de Catalunya |
| 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 |
|
| _version_ |
1869409186180235264 |
| spelling |
Rational points on Shimura curves and Galois representationsVera Piquero, Carlos deShimura curvesAtkin-Lehner quotientsRational pointsHasse principleBrauer-Manin obstructionGalois representations51This thesis explores one of the essential arithmetical and diophantine properties of Shimura curves and their Atkin-Lehner quotients: the existence of rational points on these families of curves over both number fields and their completions. Due to their moduli interpretation, Shimura curves (and modular curves) are of great arithmetic significance. The research line started by the work of Mazur on rational points on modular curves, leading to the classification of rational torsion subgroups of elliptic curves over Q, has been intensively and successfuly explored by many authors, and the general philosophy is that rational points on modular and Shimura curves over number fields should correspond only to CM-points, except for a few exceptional cases. Aiming to provide more evidence in support of this philosophy, in this thesis we propose new approaches for studying the lack of rational points over number fields on Shimura curves and their Atkin-Lehner quotients. Furthermore, we also wish to show that these curves provide a wealth of counterexamples to the Hasse principle, hence they can be used to test cohomological obstructions to this local-global principle, as for example the Brauer-Manin obstruction. The thesis is divided into two parts. The first of them is devoted to study the arithmetic and the geometry of the cyclic Galois coverings of Shimura curves introduced by Jordan. On the one hand, we determine the group of modular automorphisms of the Shimura curves arising from these coverings, showing in particular that Atkin-Lehner involutions can be lifted through them. As a consequence, we can produce cyclic étale coverings of Atkin-Lehner quotients of Shimura curves, which can be used to study the (non-)existence of rational points on these curves by applying descent techniques. Further, we characterise the existence of local points at bad reduction primes on both the intermediate curves of Jordan's coverings and their quotients by Atkin-Lehner involutions. This part of the thesis exploits the adèlic formalism of Shimura curves, as well as the padic uniformisation theory of Cerednik and Drinfeld, generalising previous work of Jordan-Livné and Ogg. In the second part of the thesis, we propose and investigate a method for proving the non-existence of rational points over a number field K on a coarse moduli space X of abelian varieties with additional structure, with special interest in cases where the moduli problem is not fine and K-rational points may not be represented by abelian varieties admitting a model over K (which is the generic situation if the abelian varieties being classified have even dimension). The original inspiration dates back to the works of Mazur and Jordan, in which the authors study the existence K-rational points on modular and Shimura curves, respectively, through the Galois representations attached to the elliptic curves and abelian surfaces parametrised by them. However, they need to assume these varieties to be defined over K (their field of moduli), a hypothesis which need not be correlated to the non-existence of K-rational points on the moduli space. To overcome this, we attach Galois representations to K-rational points on X rather than to the abelian varieties classified by them (what we call "Galois representations over fields of moduli"), inspired by the work of Ellenberg and Skinner on the modularity of Q-curves. We exemplify our method, combined with the cyclic coverings studied in the first part of the thesis, in the case where X is either a Shimura curve or an Atkin-Lehner quotient of it and K is an imaginary quadratic field or the field of rational numbers, respectively. In both cases, we produce new counterexamples to the Hasse principle. And moreover, in the first case we prove that these counterexamples are accounted for by the Brauer-Manin obstruction. The results of this second part complement previous work of Parent-Yafaev, Gillibert or Clark, for example.Aquesta tesi estudia una de les propietats aritmètiques essencials de les corbes de Shimura i els seus quocients d'Atkin-Lehner: l'existència de punts racionals en aquestes famílies de corbes sobre cossos de nombres i les seves complecions. Degut a la seva interpretació modular, les corbes de Shimura (i les corbes modulars) tenen gran interès aritmètic. La recerca iniciada per Mazur sobre punts racionals en corbes modulars, que dugué a la classificació dels subgrups racionals de torsió de corbes el·líptiques sobre Q, ha estat explorada per diversos autors. La filosofia general és que els únics punts racionals en corbes modulars i de Shimura sobre cossos de nombres corresponen a punts CM, llevat de casos excepcionals. Amb l'objectiu d'aportar nous arguments a favor d'aquesta filosofia, aquesta tesi proposa nous mètodes per estudiar l'absència de punts racionals sobre cossos de nombres en corbes de Shimura i quocients d'Atkin-Lehner. A més, també volem evidenciar que aquestes corbes proporcionen un bon nombre de contraexemples al principi de Hasse, i per tant poden servir per estudiar obstruccions cohomològiques a aquest principi local-global, com ara l'obstrucció de Brauer-Manin. La tesi està dividida en dues parts. La primera està dedicada a l'estudi de l'aritmètica i la geometria dels recobridors cíclics de Galois de corbes de Shimura introduits per Jordan. D'una banda, determinem el grup d'automorfismes modulars de les corbes que sorgeixen d'aquests recobridors, provant en particular que les involucions d'Atkin-Lehner s'aixequen a aquests recobridors. En conseqüència, construïm recobridors cíclics no ramificats de quocients d'Atkin-Lehner de corbes de Shimura, útils per estudiar l'absència de punts racionals en aquestes corbes aplicant tècniques de descens. A més, caracteritzem l'existència de punts locals en primers de mala reducció en les corbes intermitjes dels recobridors de Jordan i els seus quocients per involucions d'Atkin-Lehner. Aquesta part de la tesi explota el formalisme adèlic de les corbes de Shimura, així com la teoria d'uniformització p-àdica de Cerednik i Drinfeld, generalitzant treballs previs de Jordan-Livné i Ogg. A la segona part de la tesi, proposem i investiguem un mètode per provar l'absència de punts racionals sobre un cos de nombres K en un espai de mòduli X de varietats abelianes amb estructura addicional, amb interès especial en el cas on el problema de mòduli no és fi i punts K-racionals poden no ésser representats per varietats abelianes admetent un model sobre K (que és el cas genèric si les varietats abelianes parametritzades tenen dimensió parella). La inspiració original es remunta als treballs de Mazur i Jordan, on els autors estudien punts K-racionals en corbes modulars i de Shimura, respectivament, a través de les representacions de Galois associades a les corbes el·líptiques i superfícies abelianes que parametritzen. Tanmateix, cal suposar que aquestes varietats admeten un model sobre K (el seu cos de mòduli), hipòtesi que no té per què estar relacionada amb l'absència de punts K-racionals en l'espai de mòduli. Per superar aquesta dificultat, associem representacions de Galois a punts K-racionals en X enlloc de fer-ho a les varietats abelianes que aquests punts classifiquen (el que anomenem "representacions de Galois sobre cossos de mòduli"), inspirats pel treball d'Ellenberg i Skinner sobre la modularitat de les Q-corbes. Exemplifiquem el nostre mètode, combinat amb els recobridors de la primera part de la tesi, en el cas on X és una corba de Shimura o un quocient d'Atkin-Lehner seu, i K és un cos quadràtic imaginari o el cos dels nombres racionals, respectivament. En ambdòs casos, produïm nous contraexemples al principi de Hasse. A més, en el primer cas demostrem que aquests contraexemples són explicats per l'obstrucció de Brauer-Manin. Els resultats d'aquesta segona part complementen treballs previs de Parent-Yafaev, Gillibert o Clark, entre d'altres.DOCTORAT EN MATEMÀTICA APLICADA (Pla 2007)Universitat Politècnica de CatalunyaRotger, VíctorUniversitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II201420142014info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersion144 p.application/pdfapplication/pdfhttp://hdl.handle.net/10803/284400https://dx.doi.org/10.5821/dissertation-2117-95533TDX (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-nc-sa/3.0/es/http://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:www.tdx.cat:10803/2844002026-06-14T12:46:07Z |
| score |
15,301629 |