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

TL;DR

This paper surveys the development of Kaneko's formula, which links Fourier coefficients of the elliptic modular j-function to singular moduli, highlighting its extensions and broader implications in modular forms and functions.

Contribution

It provides a comprehensive overview of the evolution and extensions of Kaneko's formula connecting Fourier coefficients and singular moduli in modular forms.

Findings

Kaneko's formula relates Fourier coefficients to singular moduli.

Extensions broaden the connection between modular forms and special values.

The survey highlights the impact on understanding modular functions.

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.