Critical points of modular forms
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911955285966848 |
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| author | van Ittersum, Jan-Willem Ringeling, Berend |
| author_facet | van Ittersum, Jan-Willem Ringeling, Berend |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_00432 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Critical points of modular forms van Ittersum, Jan-Willem Ringeling, Berend Number Theory 11F11, 11F99 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. |
| title | Critical points of modular forms |
| topic | Number Theory 11F11, 11F99 |
| url | https://arxiv.org/abs/2204.00432 |