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# Communications in Number Theory and Physics

## Volume 7 (2013)

### Number 4

### Automorphic forms for triangle groups

Pages: 689 – 737

DOI: http://dx.doi.org/10.4310/CNTP.2013.v7.n4.a4

#### Authors

#### Abstract

For triangle groups, the (quasi-)automorphic forms are known just as explicitly as for the modular group $\mathrm{PSL}(2,\mathbb{Z})$. 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 three-folds over the thrice-punctured sphere.