  We consider the moduli space of abelian varieties with two marked points and
a frame of the relative de Rham cohomolgy 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.
