Meromorphic modular differential equations for the elliptic genus of Calabi--Yau varieties

We study differential equations – not necessarily monic – for Jacobi modular forms. As an application, we derive a second-order differential equation for the elliptic genus of a $K3$ surface involving the logarithmic derivative of the modular invariant $j(\tau )$, and we explicitly find a third-order equation with a leading coefficient of weight 12 for the elliptic genus of any four-dimensional Calabi–Yau variety. Furthermore, we propose a simple recursive algorithm for investigating second-order differential equations with leading coefficient $E_4(\tau )$ or $E_6(\tau )$.