On the group of automorphisms of Shimura curves and applications

2002/06/10 by Victor Rotger, Rotger, Victor · 1 citation
Mathematics · #11G18 #14G35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G18 #msc:14G35

paper · pdf · doi:10.48550/arxiv.math/0206101

To appear in Compositio Mathematica

arxiv created 2002/06/10 · arxiv updated 2009/11/30

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 \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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• Point counts, automorphisms, and gonalities of Shimura curves

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