A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms

Abstract

In this paper we give a realization of the Shimura curve for the quaternion algebra over Q with discriminant 6 as a quotient space of the complex upper half plane by the triangle group Δ(3, 6, 6). It is given by the Schwarz map for the Gauss hypergeometric differential equation $${E\left(\frac{1}{6},\frac{1}{3},\frac{2}{3}\right)}$$ . The corresponding abelian surfaces are obtained as isogenous components of the Jacobi varieties of the Picard curves $${C(s): w^3=z(z-1)\break \left(z-\frac{1}{2} (1-s)\right)\left(z-\frac{1}{2} (1+s)\right)}$$ .