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Quantum modular forms and Hecke operators

Seewoo Lee

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.

Quantum modular forms and Hecke operators

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.

Paper Structure

This paper contains 7 sections, 15 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modular forms and Period polynomials
  4. Dedekind symbols and correspondences
  5. Quantum modular forms
  6. Quantum modular forms associated to Dedekind symbols with polynomial reciprocity law
  7. Hecke Operators on Quantum modular forms

Key Result

Theorem 1.1

Let $w\geq 2$ be an even integer. Define the following spaces :

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Eichler-Shimura-Zagier
  • Theorem 2.2: Choie-Zagier
  • Theorem 2.3: Fukuhara
  • Theorem 2.4: Fukuhara
  • Definition 2.1
  • Proposition 2.1
  • Definition 3.1
  • Proposition 3.1
  • ...and 17 more