Quantum Modular Forms from Real Quadratic Double Sums

TL;DR

This paper explores the modular and quantum modular properties of specific $q$-hypergeometric series linked to quadratic fields, revealing they form vector-valued quantum modular forms on $ ext{Gamma}_0(2)$.

Contribution

It demonstrates that twelve series discovered by Lovejoy and Osburn are actually vector-valued quantum modular forms, expanding understanding of their structure and properties.

Findings

The series form vector-valued quantum modular forms on $ ext{Gamma}_0(2)$.

They exhibit specific modular and quantum modular properties.

The Fourier coefficients relate to counting functions of ideals in quadratic fields.

Abstract

In 2015, Lovejoy and Osburn discovered twelve $q$-hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and quantum modular properties and show that they yield three vector-valued quantum modular forms on the group $\Gamma_0 (2)$.