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

Abstract

Abstract 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.