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(cos 2π/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 Χ0 (3), Χ1 (3) associated with the congruence subgroups Γ0 (3), Γ 1(3). Since the rational prime 3 is inert in K, the covering Χ0(3)/Χ(1) has degree 28, and its Galois closure Χ(3)/Χ(1) has geometric Galois group PSL2(F27). Since Χ(1) is rational, the covering Χ0(S)/Χ(1.) amounts to a rational map of degree 28. We compute this rational map explicitly. We find that Χ0(3) is an elliptic curve of conductor 147 = 3 · 72 over ℚ, as is the Jacobian ℑ1 (3) of Χ1 (3); that these curves are related by an isogeny of degree 13; and that the kernel of the 13-isogeny from ℑ1 (3) to Χ0 (3) consists of K-rational points. We also use the map Χ0(3) → Χ(1) to locate some complex multiplication (CM) points on Χ(1). We conclude by describing analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields. © Springer-Verlag Berlin Heidelberg 2006.