Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero

TL;DR

This paper explores the algebraic structure of higher genus quasi Jacobi forms of index zero through the moduli space of abelian varieties, explicitly computing the Gauss-Manin connection for elliptic curves.

Contribution

It introduces an algebro-geometric framework linking higher genus quasi Jacobi forms to the Gauss-Manin connection and explicitly computes this connection for elliptic curves.

Findings

Explicit computation of Gauss-Manin connection for elliptic curves

Establishment of a geometric framework for higher genus quasi Jacobi forms

Description of differential equations as vector fields on moduli space

Abstract

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.