Half-integral weight Eichler integrals and quantum modular forms

TL;DR

This paper develops a general theory of quantum modular forms arising from half-integral weight Eichler integrals, expanding their mathematical understanding and highlighting their significance in combinatorics and number theory.

Contribution

It introduces a comprehensive framework for quantum modular forms linked to half-integral weight Eichler integrals, unifying previous examples and proposing new research directions.

Findings

Established a well-defined class of quantum modular forms.

Connected quantum modular forms to combinatorial and number theoretical problems.

Outlined fundamental open questions for future research.

Abstract

In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a new type of object, which Zagier later called a quantum modular form. Since then, a number of others have studied similar examples. Here we develop the theory in a general context, giving rise to a well-defined class of quantum modular forms. Since elements of this class show up frequently in examples of combinatorial and number theoretical interest, we propose the study of the general properties of this space of quantum modular forms. We conclude by giving a set of fundamental questions concerning this space of objects which merit further study.