國立清華大學 數學系

Abstract: We introduce a method for presenting explicit models of hyperelliptic Shimura curves attached to indefinite quaternion algebras over rational field and Atkin-Lehner quotients of them. It ultilize 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 problem 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. Tea Time: 3:00PM, R707