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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.14500 |
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| _version_ | 1866918026863968256 |
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| author | Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem |
| author_facet | Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem |
| contents | We prove two of Kaneko's conjectures on the "values" $\mathrm{val}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{Re}(\mathrm{val}(w))\geq \mathrm{val}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{Re}(\mathrm{val}(w))\leq \mathrm{val}\left(1+\sqrt{2}\right)$ for all Markov irrationalities $w$. These results generalize to the "values" at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [π/3,π/2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real part of cycle integrals and conjectures of Kaneko Bengoechea, Paloma Herrero, Sebastián Imamoglu, Özlem Number Theory 11F03, 11J06 We prove two of Kaneko's conjectures on the "values" $\mathrm{val}(w)$ of the modular $j$ function at real quadratic irrationalities: we prove the lower bound $\mathrm{Re}(\mathrm{val}(w))\geq \mathrm{val}\left(\frac{1+\sqrt{5}}{2}\right)$ for all real quadratics $w$ and the upper bound $\mathrm{Re}(\mathrm{val}(w))\leq \mathrm{val}\left(1+\sqrt{2}\right)$ for all Markov irrationalities $w$. These results generalize to the "values" at quadratic irrationalities of any weakly holomorphic modular function $f$ such that $f(e^{it})$ is real, non-negative and increasing for $t\in [π/3,π/2]$. |
| title | Real part of cycle integrals and conjectures of Kaneko |
| topic | Number Theory 11F03, 11J06 |
| url | https://arxiv.org/abs/2505.14500 |