Automorphic Schwarzian equations and integrals of weight 2 forms

Abstract

In this paper, we investigate the non-modular solutions to the Schwarz differential equation \(\{f,\tau \}=sE_4(\tau )\) 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 \(n\equiv 1\mod 12\). These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation \(\ \displaystyle y''+\frac{\pi ^2n^2}{36}\,E_4\,y=0\) for each \(n\equiv 1\mod 12\) 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.