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 $1\le n\le 6$.