Automorphic Schwarzian equations and integrals of weight 2 forms

TL;DR

This paper explores non-modular solutions to a specific Schwarzian differential equation involving Eisenstein series, providing explicit integral solutions for certain parameters and generalizing classical results by Hurwitz and Klein.

Contribution

It introduces explicit solutions for non-modular cases of the Schwarzian equation using integrals of weight 2 modular forms, extending previous work on modular solutions.

Findings

Explicit solutions for $s=2\pi^2(n/6)^2$ with $n ot\equiv 0 ext{ mod }12$

Generalization of classical results for $n=1$ to other values of $n$

Connection between solutions and equivariant functions on the upper half-plane

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.