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

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

2004-09-01

arxiv.org/abs/math/0409020v2

Lecture Notes in Computer Science 4076 (proceedings of ANTS-7, Berlin 2006; F.Hess, S.Pauli, and M.Pohst, ed.), 302--316

Mathematics

Number Theory

15 pages. This is the published version, considerably improved by the suggestions of the referees, one of whom identified hims

Scientific paper

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.

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