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arXiv · 2507.21514

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

Toshiki Matsusaka
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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.

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Toshiki Matsusaka. 2025-07-29. The Fourier coefficients and singular moduli of the elliptic modular function $j(\tau)$, revisited. https://arxiv.org/abs/2507.21514

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