Maass wave forms, quantum modular forms and Hecke operators

Abstract

Andrew, Dyson and Hickerson proved Andrew’s conjecture on coefficients of a q-series σ(q) by discovering connection between the coefficients and arithmetic of Q(√6). Using this, Cohen proved that the coefficients can be interpreted as coefficients of a certain Maass wave form on Γ_0(2) with a nontrivial multiplier system ν_C. Also, Zagier associated a quantum modular form to the Cohen’s Maass wave form. In this paper, using Wolhfhart’s operator, we define Hecke operators on the space of Maass wave forms and quantum modular forms which change multiplier system, and prove that the Maass wave form and the quantum modular form are eigenforms with respect to these operators. As a corollary, we find new identity of the p-th coefficients of σ(q) in terms of p-th root of unities. Also, we proved similar thing for Li-Ngo-Rhoades’ Maass wave form and the associated quantum modular form.