Singular invariants and coefficients of harmonic weak Maass forms of weight 5/2

Abstract

Abstract We study the coefficients of a natural basis for the space of harmonic weak Maass forms of weight 5 / 2 ${5/2}$ on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p ⁢ ( n ) ${p(n)}$ . We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoḡlu, and Tóth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.