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

Abstract

The (2,3,7) triangle group is known to be associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(cos2π/7) and unramified at all other places of K. This triangle group and its congruence subgroups thus give rise to various Shimura curves and maps between them. We study the genus-1 curves ${{\cal X}_0}(3)$, ${{\cal X}_1}(3)$ associated with the congruence subgroups Γ₀(3), Γ₁(3). Since the rational prime 3 is inert in K, the covering ${{\cal X}_0}(3)/{{\cal X}(1)}$ has degree 28, and its Galois closure ${\cal X}(3)/{{\cal X}(1)}$ has geometric Galois group PSL₂(F ₂₇). Since ${{\cal X}(1)}$ is rational, the covering ${{\cal X}_0}(3)/{{\cal X}(1)}$ amounts to a rational map of degree 28. We compute this rational map explicitly. We find that ${{\cal X}_0}(3)$ is an elliptic curve of conductor 147=37² over Q, as is the Jacobian ${{\cal J}_1}(3)$ of ${{\cal X}_1}(3)$; that these curves are related by an isogeny of degree 13; and that the kernel of the 13-isogeny from ${{\cal J}_1}(3)$ to ${{\cal X}_0}(3)$ consists of K-rational points. We also use the map ${{\cal X}_0}(3) \rightarrow {{{\cal X}}(1)}$ to locate some complex multiplication (CM) points on ${{\cal X}(1)}$. We conclude by describing analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields.