Non-elliptic Shimura curves of genus one

Abstract

We present explicit models for non-elliptic genus one Shimura curves $X_0(D,N)$ with ${\mathrm{\Gamma }}_0\left(N\right)$-level structure arising from an indefinite quaternion algebra of reduced discriminant $D$, and Atkin-Lehner quotients of them. In addition, we discuss and extend Jordan's work [10, Ch. III] on points with complex multiplication on Shimura curves.