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arXiv · 2204.00432

Critical points of modular forms

Jan-Willem van Ittersum·Berend Ringeling
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Abstract

We count the number of critical points of a modular form with real Fourier coefficients in a $γ$-translate of the standard fundamental domain $\mathcal{F}$ (with $γ\in \mathrm{SL}_2(\mathbb{Z})$). Whereas by the valence formula the (weighted) number of zeros of this modular form in $γ\mathcal{F}$ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of $\mathcal{F},$ the value of $γ^{-1}(\infty)$ and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form.

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BibTeXRIS

Jan-Willem van Ittersum, Berend Ringeling. 2024-07-15. Critical points of modular forms. https://doi.org/10.1142/s1793042124501288

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