Symmetric tensor representations,quasimodular forms, and weak Jacobi forms

TL;DR

This paper establishes a mathematical correspondence linking vector-valued modular forms, quasimodular forms, and weak Jacobi forms through explicit isomorphisms, extending classical results with new algebraic structures.

Contribution

It introduces an explicit isomorphism connecting vector-valued modular forms with symmetric tensor representations to sequences of modular forms, extending prior work by Kuga and Shimura.

Findings

Established a correspondence between vector-valued modular forms and quasimodular forms.

Derived an explicit isomorphism using Rankin-Cohen brackets.

Connected vector-valued modular forms to weak Jacobi forms.

Abstract

We establish a correspondence between vector-valued modular forms with respect to a symmetric tensor representation and quasimodular forms. This is carried out by first obtaining an explicit isomorphism between the space of vector-valued modular forms with respect to a symmetric tensor representation and the space of finite sequences of modular forms of certain type. This isomorphism uses Rankin-Cohen brackets and extends a result of Kuga and Shimura, who considered the case of vector-valued modular forms of weight two. We also obtain a correspondence between such vector-valued modular forms and weak Jacobi forms.