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.

arXiv comment25 pages. This survey is based on a talk given at the conference "Modular Forms and Multiple Zeta Values -Conference in Honor of Masanobu Kaneko's 60+4th Birthday", held at Kindai University in February 2025