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

TL;DR

This paper develops methods to compute equations for Shimura curves using K3 surfaces with high Picard number, enabling the study of these curves and CM points beyond previous capabilities.

Contribution

It introduces a novel approach leveraging K3 surfaces with Picard number at least 19 to compute Shimura curves and CM points more effectively.

Findings

Successfully computed equations for various Shimura curves.

Mapped natural relationships between Shimura curves.

Identified CM points on these curves with high precision.

Abstract

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) = <http://arxiv.org/abs/math/0005160>