Periods of the j-function along infinite geodesics and mock modular forms

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Zagier's well-known work on traces of singular moduli relates the coefficients of certain weakly holomorphic modular forms of weight 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 1/2 mock modular forms fD = ∑d>₀ a(d,D) qᵈ, 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 jₘ,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 jₘ,Q.