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Uniformization of certain Shimura curves

Pilar Bayer — 2002

Banach Center Publications

We present an approach to the uniformization of certain Shimura curves by means of automorphic functions, obtained by integration of non-linear differential equations. The method takes as its starting point a differential construction of the modular j-function, first worked out by R. Dedekind in 1877, and makes use of a differential operator of the third order, introduced by H. A. Schwarz in 1873.

On equations defining fake elliptic curves

Pilar BayerJordi Guàrdia — 2005

Journal de Théorie des Nombres de Bordeaux

Shimura curves associated to rational nonsplit quaternion algebras are coarse moduli spaces for principally polarized abelian surfaces endowed with quaternionic multiplication. These objects are also known as . We present a method for computing equations for genus 2 curves whose Jacobian is a fake elliptic curve with complex multiplication. The method is based on the explicit knowledge of the normalized period matrices and on the use of theta functions with characteristics. As in the case of CM-points...

Uniformization of triangle modular curves.

Pilar BayerArtur Travesa — 2007

Publicacions Matemàtiques

In the present article, we determine explicit uniformizations of modular curves attached to triangle Fuchsian groups with cusps. Their Hauptmoduln are obtained by integration of non-linear differential equations of the third order. Series expansions involving integral coefficients are calculated around the cusps as well as around the elliptic points. The method is an updated form of a differential construction of the elliptic modular function j, first performed by Dedekind...

Quadratic modular symbols on Shimura curves

Pilar BayerIván Blanco-Chacón — 2013

Journal de Théorie des Nombres de Bordeaux

We introduce the concept of modular symbol and study how these symbols are related to p -adic L -functions. These objects were introduced in [] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p -adic L -functions to more general Shimura curves.

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