Equivariant solutions to modular Schwarzian equations

TL;DR

This paper provides explicit solutions to a class of modular Schwarzian differential equations for all positive integers r, using equivariant functions, and characterizes their modular properties depending on the parity of r.

Contribution

It extends previous solutions for specific rational r to all positive integers, offering a comprehensive method to solve these equations via equivariant functions.

Findings

Solutions expressed in terms of equivariant functions.

Solutions are quasi-modular forms for SL_2(Z) or its subgroup.

Provides explicit solutions for all positive integer r.

Abstract

For every positive integer $r$, we solve the modular Schwarzian differential equation $\{h,\tau\}=2\pi^2r^2E_4$, 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 $y''+r^2\pi^2E_4\,y=0$ 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.