Traces of Poincaré series at square discriminants and Fourier coefficients of mock modular forms
Abstract
Jeon, Kang and Kim (2013) proved that the Fourier coefficients $b(d,D)$ of mock modular form $\mathscr G_{d}$ of weight $3/2$ for $\Gamma_{0}(4)$ can be interpreted in terms of the Hurwitz–Kronecker class number and modified traces of cycle integrals of a sesqui-harmonic Maass form $\hat{\mathbb J}_{1}$ of weight $0$ for ${\rm SL}_{2}(\mathbb Z)$. The function $\hat{\mathbb J}_{1}$ is related to Klein’s $j$-function via its image under hyperbolic Laplacian. Their result applies to the coefficients $b(d,D)$ for which $dD$ is not a square. In this paper, we define modified traces for square discriminants and express the coefficients $b(d, D)$ with $dD$ a square in terms of the Hurwitz–Kronecker class number and modified traces of cycle integrals of $\hat{\mathbb J}_{1}$. Furthermore, we prove that the coefficient $b(d, D)$ is the regularized inner product of weakly holomorphic modular forms of weight $1/2$ for $\Gamma_{0}(4)$. As an application, we express the modified trace of $\hat{\mathbb J}_{1}$ at a square discriminant in terms of the central critical value of the (non-existent) $L$-series of harmonic Maass form of ‘dual’ weight. The summands that emerge in the expression of the central critical value are linked to a classical expression of the Rademacher–Petersson type formula.
