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

andersen, nickolas;

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, Imamo¯glu, and T´oth. They showed that the coefficients of certain weight 1/2 mock modular forms
fD =d>0a(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 Imamo¯glu
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, Imamo¯glu, and T´oth, that the coefficients for which dD is a square are traces of cycle integrals of the functions jm,Q.