Automorphic Schwarzian equations

Abstract

This paper concerns the study of the Schwarz differential equation $\{h,\tau \}=s{E}_4(\tau )$ where $E_4$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we determine all the values of $s$ for which the solutions $h$ are modular functions for a finite index subgroup of $\mbox{SL}_2(\mathbb Z)$. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of $\mbox{SL}_2(\mathbb Z)$. This also leads to the solutions to the Fuchsian differential equation $y^{\prime \prime }+s{E}_{4}y=0$.