On the Group of Automorphisms of Shimura Curves and Applications

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. 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.