Hypergeometric solutions to Schwarzian equations

Abstract

In this paper we study the modular differential equation \(y''+s\,E_4\, y=0\) where \(E_4\) is the weight 4 Eisenstein series and \(s=\pi ^2r^2\) with \(r=n/m\) being a rational number in reduced form such that \(m\ge 7\). This study is carried out by solving the associated Schwarzian equation \(\{h,\tau \}=2\,s\,E_4\) and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases \(1\le m\le 6\) have already been treated in Saber and Sebbar (Forum Math 32(6):1621–1636, 2020; Ramanujan J 57(2):551–568, 2022; J Math Anal Appl 508:125887, 2022; Modular differential equations and algebraic systems, http://arxiv.org/abs/2302.13459).