2018-06-21T20:10:42Zhttps://www.tdx.cat/oai/requestoai:www.tdx.cat:10803/4022092017-09-19T07:33:08Zcom_10803_1col_10803_2
Shimura curves and their p-adic uniformization = Corbes de Shimura i les seves uniformitzacions p-àdiques
Milione, Piermarco
piermarcomilione@gmail.com
false
Bayer i Isant, Pilar, 1946-
true
Corbes algebraiques
Curvas algebraicas
Algebraic curves
Geometria algebraica
Geometría algebraica
Algebraic geometry
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.
2017-04-11T12:03:37Z
2017-04-11T12:03:37Z
2016-01-29
info:eu-repo/semantics/doctoralThesis
info:eu-repo/semantics/publishedVersion
http://hdl.handle.net/10803/402209
eng
L'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/
info:eu-repo/semantics/openAccess
Universitat de Barcelona
TDX (Tesis Doctorals en Xarxa)