Equivariant solutions to modular Schwarzian equations

Abstract

For every positive integer r, we solve the modular Schwarzian differential equation {h,τ}=2π2r2E4, where E4 is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper supplements previous works by the authors [20,21], 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″+r2π2E4y=0 for every positive integer r. These solutions are quasi-modular forms for SL2(Z) if r is even or for the subgroup of index 2, SL2(Z)2, if r is odd.