Real part of cycle integrals and conjectures of Kaneko

TL;DR

This paper proves two conjectures by Kaneko regarding bounds on the real parts of modular function values at real quadratic irrationalities, extending results to certain weakly holomorphic modular functions.

Contribution

It establishes bounds on the real parts of modular function values at quadratic irrationals, confirming Kaneko's conjectures and generalizing to specific weakly holomorphic functions.

Findings

Proved lower bound for real parts of val(w) at quadratic irrationals.

Proved upper bound for real parts of val(w) at Markov irrationalities.

Extended results to certain weakly holomorphic modular functions.

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