Arithmetic formulas for the coefficients of the McKay-Thompson series

By the work of R. Borcherds we know the Fourier coefficients of the elliptic modular $j$-function are closely related
to the Monster group. As a new perspective, M. Kaneko gave an arithmetic formula for the Fourier coefficients expressed
in terms of the traces of the CM values of the $j$-function in 1996. In this talk, we consider analogues of this formula for
the McKay-Thompson series. Our main result is an explicit formula for the coefficients expressed in terms of the CM
values of certain hauptmoduln. Furthermore, we give some applications.