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arXiv · 2505.14500

Real part of cycle integrals and conjectures of Kaneko

Paloma Bengoechea·Sebastián Herrero·Özlem Imamoglu
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Abstract

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 $t\in [\pi/3,\pi/2]$.

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Paloma Bengoechea, Sebastián Herrero, Özlem Imamoglu. 2025-05-20. Real part of cycle integrals and conjectures of Kaneko. https://arxiv.org/abs/2505.14500

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