Automorphic forms for triangle groups

TL;DR

This paper explicitly describes automorphic forms for triangle groups, explores their arithmetic properties, and connects them to Calabi-Yau threefold periods, extending classical modular form theory to new geometric contexts.

Contribution

It provides explicit formulas for automorphic forms of triangle groups and links their coefficients' arithmetic properties to the groups' arithmeticity.

Findings

Explicit expressions for automorphic forms of triangle groups

Arithmeticity of Fourier and Taylor coefficients analyzed

Modular interpretation of Calabi-Yau periods established

Abstract

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.