On the group of automorphisms of Shimura curves and applications

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.