Eisenstein series and modular differential equations (Q2888794)

Let \(E_4(z)\) be the Eisenstein series of weight \(4\). In this article, the authors give solutions of the Riccati equation:~\((k/i\pi)u'+u^2=E_4(z)\), the corresponding second order equation:~\(y''+(\pi^2/k^2)E_4(z)y=0\) and the Schwarzian differential equation:~\(\{f,z\}=(4\pi^2/k^2)E_4(z)\), where \(k\in\mathbb Z\),~\(1\leq k\leq 6\) and \(\{f,z\}=2(f''/f')'-(f''/f')^2\). If a solution to one of theses three equations with the same \(k\) is known, then solutions to two other equations can be obtained from it. In the case \(k=6\), by the Ramanujan identities for Eisenstein series, the authors show that the Riccati equation has a solution \(u=-E_2(z)=-(2\pi i)^{-1}\Delta'/\Delta\) and the corresponding second order equation has a solution \(y=\Delta^{-\frac 1{12}}\), where \(\Delta\) is the modular discriminant. In other cases, they use the Schwarzian equation to find solutions. They show in the cases \(2\leq k\leq 5\) that any Hauptmodul for the principal congruence subgroup \(\Gamma(k)\) of level \(k\) is a solution to the Schwarzian equation, and in the case \(k=1\) that \(f(z)=z+4E_4/E_4'\) is a solution to the Schwarzian equation and further \(f(z)\) is an equivariant form with respect to \(\text{SL}_2(\mathbb Z)\), thus, \(f(z)\) satisfies \(f(\gamma(z))=\gamma(f(z)), \text{for all } \gamma\in\text{SL}_2(\mathbb Z)\).