Shimura curves for level-3 subgroups of the (2,3,7) triangle group, and some other examples

Abstract

We determine the Shimura modular curve X_0(3) and the Jacobian of the Shimura modular curve X_1(3) associated with the congruence subgroups Gamma_0(3), Gamma_1(3) of the (2,3,7) triangle group. This group is known to be arithmetic and associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(\cos 2\pi/7) and at no finite primes of K. Since the rational prime 3 is inert in K, the covering X_0(3)/X(1) has degree 28 and Galois group PSL_2(F_27). We determine X_0(3) by computing this cover. We find that X_0(3) is an elliptic curve of conductor 147=3*7^2 over Q, as is the Jacobian J_1(3) of X_1(3); that these curves are related by an isogeny of degree (27-1)/2=13; and that the kernel of the 13-isogeny from J_1(3) to X_0(3) consists of K-rational points. We explain these properties, use X_0(3) to locate some complex multiplication (CM) points on X(1), and describe analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields.