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

2025/07/29 by Matsusaka, Toshiki
#11F30 #11F37 #11F50 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.21514

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.

Citations

• Hecke equivariance of the divisor map
• A unified approach to Rohrlich-type divisor sums
• Eichler-Selberg relations for singular moduli
• The arithmetic of modular grids
• Traces of CM values and cycle integrals of polyharmonic Maass forms
• Polyharmonic weak Maass forms of higher depth for SL₂(Z)
• Shintani theta lifts of harmonic Maass forms
• Shintani theta lifts of harmonic Maass forms
• The Fourier coefficients of the McKay-Thompson series and the traces of CM values
• On Divisors of Modular Forms
• On a theta lift related to the Shintani lift
• Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5
• Regularized theta liftings and periods of modular functions
• Multiple Dirichlet Series and Shifted Convolutions
• Cycle integrals of the j-function and mock modular forms
• Observations on the "values" of the elliptic modular function j(tau) at real quadratics
• Traces of CM values of modular functions
• Traces of singular values and Borcherds products
• Borcherds products associated with certain Thompson series
• Automorphic forms onO s +2,2(R) and infinite products

Related

• Exact formulas for traces of singular moduli of higher level modular functions
• The Fourier coefficients of the McKay-Thompson series and the traces of CM values
• An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular j-function
• Observations on the "values" of the elliptic modular function j(tau) at real quadratics
• Elliptic curves and Fourier coefficients of meromorphic modular forms