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

Abstract

Let $\Gamma \subset PSL_2({\mathbb R})$ 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 ${\mathfrak{H}}$; the quotient $X_{\mathbb C}=\Gamma \backslash {\mathfrak{H}}$ is a Shimura curve, and there is a map $j:X_{\mathbb C} \to {\mathbb P}^1_{\mathbb C}$. We algorithmically apply the Shimura reciprocity law to compute CM points $j(z_D) \in {\mathbb P}^1_{\mathbb C}$ 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.