Shimura Curves for Level-3 Subgroups of the (2, 3, 7) Triangle Group, and Some Other Examples

We determine the Shimura modular curve X0(3) and the Jacobian of the Shimura modular curve X1(3) associated with the congruence subgroups Γ0(3), Γ1(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 2π/7) and at no finite primes of K. Since the rational prime 3 is inert in K, the covering X0(3)/X (1) has degree 28 and Galois group PSL2(F27). We determine X0(3) by computing this cover. We find that X0(3) is an elliptic curve of conductor 147 = 3 · 7 2 over Q, as is the Jacobian J1(3) of X1(3); that these curves are related by an isogeny of degree (27 − 1)/2 = 13; and that the kernel of the 13-isogeny from J1(3) to X0(3) consists of K-rational points. We explain these properties, use X0(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. Let K be the totally real cubic field Q(cos 2π/7) of minimal discriminant (namely discriminant 49), and let A be a quaternion algebra over K ramified at two of the three real places and at no finite place of K. Fix a maximal order O ⊂ A. From the general theory of quaternion algebras over number fields (see [Vi] for instance), we know that such A exists and is determined uniquely up to isomor-phism, and that O is determined uniquely up to conjugation in A (this since A has an unramified real place and is thus indefinite). Moreover, since A has exactly one unramified real place, the group Γ(1) of elements of O of (reduced) norm 1 embeds into SL 2 (R) as a discrete co-compact subgroup. Let H be the upper half-plane, with the usual action of PSL 2 (R) = SL 2 (R)/{±1}. Then Γ(1)/{±1} acts discretely and co-compactly on H, the quotient being a Shimura 1