Real part of cycle integrals and conjectures of Kaneko

TL;DR

This work resolves Kaneko’s conjectures on the real part of the modular j-valuation at quadratic irrationalities by expressing $\mathrm{Re}(\mathrm{val}_f(w))$ as a GL$(2,\mathbb{Z})$-invariant cycle-integral against a symmetrized kernel. The authors derive an explicit integral formula for $\mathrm{Re}(\mathrm{val}_f(w))$ in terms of auxiliary sums $\hat{S}(w,t)$ and prove two sharp bounds: $\mathrm{Re}(\mathrm{val}_f(w)) \ge \mathrm{val}_f(\phi)$ for all real quadratic $w$ and $\mathrm{Re}(\mathrm{val}_f(w)) \le \mathrm{val}_f(\psi)$ for Markov irrationalities, with $\phi=\frac{1+\sqrt{5}}{2}$ and $\psi=1+\sqrt{2}$. The proofs proceed via a four-step program for each theorem, introducing the concepts of good/bad terms, the auxiliary function $Z(x,t)$, and a strategic rearrangement using the GL$(2,\mathbb{Z})$-generators $\Phi$ and $\Psi$, all supported by detailed analytic lemmas in the appendix. The results extend to any weakly holomorphic modular function $f$ with $f(e^{it})$ real, nonnegative, and increasing on $[\pi/3,\pi/2]$, thereby broadening Kaneko’s conjectures to a wider class of modular objects. These bounds contribute to the understanding of Diophantine properties of modular-values and may inform future work on the imaginary part bounds as well.

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 $t\in [π/3,π/2]$.

Figures (11)

• Figure 1: The Markov tree.
• Figure 2: Plots of $\hat{S}(\phi,t)$ and $\frac{1}{2}(\hat{S}(\phi,t)+\hat{S}(\phi,\pi-t))$ for $t\in [\pi/3,2\pi/3]$.
• Figure 3: Plot of $D_{\phi,w}(t)$ for $t\in [\pi/3,2\pi/3]$ when $w=\frac{1+\sqrt{3}}{2}=[\overline{1;2}]$.
• Figure 4: Plots of $x\mapsto Z(x,t)$ for $t\in \{\pi/3,5\pi/12,\pi/2\}$.
• Figure 5: Plots of $\frac{2L(\phi)}{\cos(t)^2-1}$ and $\log(3)P(\phi,t)$ for $t\in [\pi/3,\pi/2]$.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Proposition 3.2
• Remark 3.3
• proof : Proof of Proposition \ref{['prop:formula_re_valf']}
• Example 3.4
• Lemma 4.1