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On the Group of Automorphisms of Shimura Curves and Applications

Published online by Cambridge University Press:  04 December 2007

Victor Rotger
Victor Rotger
Affiliation:
Departament d'Àlgebra i Geometria, Universitat de Barcelona, Gran Via, 585, E-08007, Barcelona, Spain. E-mail: vrotger@mat.ub.es
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Abstract

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Let VD be the Shimura curve over $\open Q$ attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of VD and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over $\bar\open Q$ and over $\open Q$ and we use it to study the set of rational points on VD over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of VD.

Keywords

Shimura curvesautomorphismsrational points
Type
Research Article
Information
Compositio Mathematica , Volume 132 , Issue 2 , June 2002 , pp. 229 - 241
Copyright
© 2002 Kluwer Academic Publishers