On the non-existence of exceptional automorphisms on Shimura curves

Abstract

We study the group of automorphisms of Shimura curves X0(D, N) attached to an Eichler order of square-free level N in an indefinite rational quaternion algebra of discriminant D>1. We prove that, when the genus g of the curve is greater than or equal to 2, Aut (X0(D, N)) is a 2-elementary abelian group which contains the group of Atkin–Lehner involutions W0(D, N) as a subgroup of index 1 or 2. It is conjectured that Aut (X0(D, N))=W0(D, N) except for finitely many values of (D, N) and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik–Drinfeld theory on their rigid analytic uniformization at primes p| D.