Hypergeometric solutions to Schwarzian equations

Abstract

In this paper we study the modular differential equation $y^{\prime \prime }+s{E}_{4}y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=\ddot{I}{€}^2{r}^2$ with $r=n/m$ being a rational number in reduced form such that $m\ge 7$. This study is carried out by solving the associated Schwarzian equation $\{h,\ddot{I}„\}=2s{E}_4$ and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases $1\le m\le 6$ have already been treated in the litterature.

Minor corrections have been made