Hypergeometric solutions to Schwarzian equations

Abstract: In this paper we study the modular differential equation $y&#39;&#39;+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=\pi<sup>2r<sup>2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out by solving the associated Schwarzian equation ${h,\tau}=2\,s\,E_4$ 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 $1\leq m\leq 6$ have already been treated in the litterature.