Schwarzian differential equations and Hecke eigenforms on Shimura curves

Abstract: Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{{10}\cdot3^{3\cdot5}{11^4})=\sqrt6}} \sqrt[6]{\frac{11}{5^5}}$$ and other similar identities.