On the real zeros of depth 1 quasimodular forms
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916862457020416 |
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| author | Im, Bo-Hae Lee, Wonwoong |
| author_facet | Im, Bo-Hae Lee, Wonwoong |
| contents | We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. In particular, we consider the derivatives of the unique weight $k$ modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant $1$. Our main results state that (1) every zero of a depth $1$ quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the real zeros of depth 1 quasimodular forms Im, Bo-Hae Lee, Wonwoong Number Theory 11F11, 11F99 We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. In particular, we consider the derivatives of the unique weight $k$ modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant $1$. Our main results state that (1) every zero of a depth $1$ quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$. |
| title | On the real zeros of depth 1 quasimodular forms |
| topic | Number Theory 11F11, 11F99 |
| url | https://arxiv.org/abs/2401.01000 |