Gauss-Manin Connection in Disguise: Quasi Jacobi Forms of Index Zero

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomology with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi-Jacobi forms of index zero and their differential equations, which are given as vector fields. In the case of elliptic curves, we compute explicitly the Gauss-Manin connection and such vector fields.