Jacobi forms and modular differential equations

The Serre derivative is a differential operator that maps modular forms of weight $k$ to modular forms of weight $k+2$. One can study differential equations with respect to this differential operator. Some examples of such equations are the Ramanujan system of differential equations and the Kaneko-Zagier equation. A similar construction takes place in the case of Jacobi forms. In my talk I will discuss differential equations of Jacobi forms and some applications related to the elliptic genus of Calabi-Yau manifolds. This is joint work with Valery Gritsenko.