Hypergeometric solutions to Schwarzian equations

In this paper we study the modular differential equation y''+s E₄\, y=0 where E₄ is the weight 4 Eisenstein series and s=-r² with $r=m/n$ being a rational number in reduced form such that m≥ 7. This study is carried out by solving the associated Schwarzian equation ,τ\=2\,s E₄ 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≤ m≤ 6 have already been treated in the litterature.