Quantum Modular Forms from Real Quadratic Double Sums

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 ${\Gamma }_0(2)$.

26 pages; v2: Brief comments added. To appear in the Quarterly Journal of Mathematics. v3: Incomplete funding information corrected