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

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

2008-02-10

arxiv.org/abs/0802.1301v1

Mathematics

Number Theory

16 pages (1 figure drawn with the LaTeX picture environment); To appear in the proceedings of ANTS-VIII, Banff, May 2008

Scientific paper

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

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