Shimura Curves Lecture 10: Quaternionic Moduli

Let B/Q be an indefinite quaternion algebra of discriminant D (we allow the case of D = 1, i.e., B ∼= M2(Q)) and ON a level N Eichler order in B. By taking Γ(ON ) = Γ(B,ON ) to be the elements of O× of positive reduced norm, we get an arithmetic Fuchsian group and hence a Riemann surface ON\H. So as not to prejudice matters, let us temporarily denote the Shimura curve associated to any quaternion order O by X(O).2 Let N be a positive integer which is prime to D. By definition a level N Eichler order is the intersection of two maximal orders, ON = O ∩O′. Here N can be characterized in any of the following ways: