On the group of automorphisms of Shimura curves and applications

Details On the group of automorphisms of Shimura curves and applications On the group of automorphisms of Shimura curves and applications

2002-06-10

arxiv.org/abs/math/0206101v1

Mathematics

Algebraic Geometry

To appear in Compositio Mathematica

Scientific paper

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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