The Fourier coefficients and singular moduli of the elliptic modular function $j(\tau)$, revisited

Abstract

Kaneko's formula expresses the Fourier coefficients of the elliptic modular $j$-function as finite sums of singular moduli. First published as a short article in 1996, it was presented as a consequence of Zagier's work inspired by Borcherds products. Since then, the formula has developed into a broader framework that links the Fourier coefficients of modular forms to the special values of modular functions, extending in various directions. This article surveys these subsequent developments.