Limits of traces of singular moduli

Let f and g be weakly holomorphic modular functions on Γ0(N) with the trivial character. For an integer d, let Trd(f) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ0(4N). In this paper, we prove that when z goes radially to r, the limit QH (f)(r) of the sum H(f)(z) =Σd>0 Trd(f)e2πidz is a special value of a regularized twisted L-function defined by Trd(f) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QH(f)(r) = QH (g)(r) for all r equivalent to i∞ under the action of Γ0(4N), then Trd(f) = Trd(g) for all integers d.