Modular differential equations and algebraic systems

Abstract

This paper investigates the modular differential equation \(y''+sE_4y=0\) on the upper half-plane \({{\mathbb {H}}}\), where \(E_4\) is the weight 4 Eisenstein series and s is a complex parameter. This is equivalent to studying the Schwarz differential equation \(\{h,\tau \}=2sE_4\), where the unknown h is a meromorphic function on \({{\mathbb {H}}}\). On the other hand, such a solution h must satisfy \(h(\gamma \tau )=\varrho (\gamma )h(\tau )\) for \(\tau \in {{\mathbb {H}}}\), \(\gamma \in {\text{ SL}_2({{\mathbb {Z}}})}\) and \(\varrho \) being a \(2-\)dimensional complex representation of \({\text{ SL}_2({{\mathbb {Z}}})}\). Moreover, in order for h to be meromorphic or to have logarithmic singularities at the \({\text{ SL}_2({{\mathbb {Z}}})}\)-cusps of \({{\mathbb {H}}}\), it is necessary to have \(s=\pi ^2r^2\) with r being a rational number. For \(r=m/n\) in reduced form, it turns out that \(\varrho \) is irreducible with finite image if and only if \(2\le n\le 5\) and in this case h is a modular function for the genus zero torsion-free principal congruence group \({\Gamma }(n)\), while\(\varrho \) is reducible if and only if \(n=6\). By Solving an explicit algebraic system, we prove that solutions for any \(r=m/n\) can be built from a solution corresponding to \(r=1/n\), for \(2\le n\le 6\), by integrating certain weight 2 meromorphic modular forms. Together with the earlier work by the authors for r being an integer [20], this provides the solutions to the above-mentioned differential equations for all \(r=m/n\) with \(1\le n\le 6\).