Computing CM Points on Shimura Curves Arising from Cocompact Arithmetic Triangle Groups

摘要

Let Γ is contained in PS L_2(R) be a cocompact arithmetic triangle group, i.e. a Fuchsian triangle group that arises from the unit group of a quaternion algebra over a totally real number field. The group Γ acts on the upper half-plane η; the quotient X_c = Γη is a Shimura curve, and there is a map j : X_c → P_c~1. We algorithmically apply the Shimura reciprocity law to compute CM points j(z_D) ∈ P_c~1 and their Galois conjugates so as to recognize them as purported algebraic numbers. We conclude by giving some examples of how this method works in practice.