Publication Date:
2015-05-19
Description:
Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight $\frac {1}{2}$ to traces of values of the modular $j$ -function at imaginary quadratic points. A real quadratic analogue was recently studied by Duke, Imamoglu, and Tóth. They showed that the coefficients of certain weight $\frac {1}{2}$ mock modular forms \[f_D = \sum _{d 〉 0} a(d,D) q^d, \quad D 〉 0\] are given in terms of traces of cycle integrals of the $j$ -function. Their result applies to those coefficients $a(d,D)$ for which $dD$ is not a square. Recently, Bruinier, Funke, and Imamoglu employed a regularized theta lift to show that the coefficients $a(d,D)$ for square $dD$ are traces of regularized integrals of the $j$ -function. In the present paper, we provide an alternate approach to this problem. We introduce functions $j_{m,Q}$ (for $Q$ a quadratic form) which are related to the $j$ -function and show, by modifying the method of Duke, Imamoglu, and Tóth, that the coefficients for which $dD$ is a square are traces of cycle integrals of the functions $j_{m,Q}$ .
Print ISSN:
0024-6093
Electronic ISSN:
1469-2120
Topics:
Mathematics
