Equivariant solutions to modular Schwarzian equations

Abstract

For every positive integer $r$, we solve the modular Schwarzian differential equation $\{h,\tau \}=2{\pi }^2{r}^2{E}_4$, where $E_4$ is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper supplements previous works \cite{forum, ramanujan}, where the same equation has been solved for infinite families of rational values of $r$. This also leads to the solutions to the modular differential equation $y^{\prime \prime }+r^2{\pi }^2{E}_{4}y=0$ for every positive integer $r$. These solutions are quasi-modular forms for $\mbox{SL}_2(\mathbb Z)$ if $r$ is even or for the subgroup of index 2, $\mbox{SL}_2(\mathbb Z)^2$, if $r$ is odd.

19 pages