# Quantum modular forms and Hecke operators

## TL;DR

This paper extends the theory of modular forms by introducing quantum modular forms with polynomial period functions, exploring their relation to existing spaces, and defining Hecke operators to generate new forms.

## Contribution

It introduces the space of quantum modular forms with polynomial period functions and develops Hecke operators acting on this space, expanding the framework of modular form theory.

## Key findings

- Established a correspondence between quantum modular forms and classical spaces
- Constructed new quantum modular forms using Hecke operators
- Extended Fukuhara's results to include quantum modular forms

## Abstract

It is known that there is an one-to-one correspondence among the space of cusp forms, the space of homogeneous period polynomials and the space of Dedekind symbols with polynomial reciprocity laws. We add one more space, the space of quantum modular forms with polynomial period functions, to extend results from Fukuhara. Also, we consider Hecke operators on the space of quantum modular forms and construct new quantum modular forms.