Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19

摘要

It is known that K3 surfaces S whose Picard number rho (= rank of theNeron-Severi group of S) is at least 19 are parametrized by modular curves X,and these modular curves X include various Shimura modular curves associatedwith congruence subgroups of quaternion algebras over Q. In a family of such K3surfaces, a surface has rho=20 if and only if it corresponds to a CM point onX. We use this to compute equations for Shimura curves, natural maps betweenthem, and CM coordinates well beyond what could be done by working with thecurves directly as we did in ``Shimura Curve Computations'' (1998) =