Special points on products of two Shimura curves

Abstract

Let S-1 and S-2 be two Shimura curves over Q attached to rational indefinite quaternion algebras B-1Q and B-2Q with maximal orders B-1 and B-2 respectively. We consider an irreducible closed algebraic curve C in the product (S-1 x S-2)(C) such that C(C) boolean AND (S-1 x S-2)(C) contains infinitely many complex multiplication points. We prove, assuming the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields. that C is of Hedge type.