Abstract
We introduce a method for presenting explicit models of hyperelliptic Shimura curves attached to indefinite quaternion algebras over the rational field and Atkin-Lehner quotients of them. It utilizes Borcherds forms, Schofer's norm formula, Kudla-Yang's formula for Whittaker functions, eta products and the realization of modular forms on Atkin-Lehner quotient of Shimura curves as Borcherds forms. The solvability of integer programming problems fatefully make us to produce sufficiently many eta products, which makes it possible to manufacture Borcherds forms with desired divisors in practice. Furthermore, combining with Shimura reciprocity law and explicit covers between Shimura curves, we could determine defining equations of hyperelliptic Shimura curves and coordinates of CM-points on these curves. We work out several examples and provide a list of equations of Shimura curves and coordinates of CM-points on these curves obtained by our method. Some of these equations are known by the works of many authors in the past decade, and about half of them are new.