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On the group of automorphisms of Shimura curves and applications

Victor Rotger
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Abstract

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

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Victor Rotger. 2002-06-10. On the group of automorphisms of Shimura curves and applications. https://arxiv.org/abs/math/0206101

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