Automorphic Schwarzian equations and integrals of weight 2 forms

Abstract

In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=s{E}_4(\tau )$ where $E_4(\tau )$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2{\pi }^2(n/6{)}^2$ with $n\equiv 1\mathrm{mod}12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $y^{\prime \prime }+\frac{{\pi }^2{n}^2}{36}{E}_{4}y=0$ for each $n\equiv 1\mathrm{mod}12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution.

20 pages