Automorphic forms for triangle groups

Understand

For triangle groups, the (quasi-)automorphic forms are known just as explicitly as for the modular group SL$(2,\bbZ)$.

• We collect these expressions here, and then interpret them using the Halphen differential equation.
• We study the arithmetic properties of their Fourier coefficients at cusps and Taylor coefficients at elliptic fixed-points --- in both cases integrality is related to the arithmeticity of the triangle group.
• As an application of our formulas, we provide an explicit modular interpretation of periods of 14 families of Calabi-Yau threefolds over the thrice-punctured sphere.