A conjecture of Zagier and the value distribution of quantum modular forms

Abstract

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when $J_{K,0}$ denotes the colored Jones polynomial of a knot $K$, Zagier's modularity conjecture describes the asymptotics of the quotient $J_{K,0} (e^{2 πi γ(x)}) / J_{K,0}(e^{2 πi x})$ as $x \to \infty$ along rationals with bounded denominators, where $γ\in \mathrm{SL}(2,\mathbb{Z})$. This problem is most accessible for the figure-eight knot $4_1$, where the colored Jones polynomial has a simple explicit expression in terms of the $q$-Pochhammer symbol. Zagier also conjectured that the function $h(x) = \log (J_{4_1,0} (e^{2 πi x}) / J_{4_1,0}(e^{2 πi /x}))$ can be extended to a function on $\mathbb{R}$ which is continuous at irrationals. In the present paper, we prove Zagier's continuity conjecture for all irrationals for which the sequence of partial quotients in the continued fraction expansion is unbounded. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of $h$, uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of $\log J_{4_1,0}(e^{2 πi x})$, when $x$ ranges over all reduced rationals in $(0,1)$ with denominator at most $N$, as $N \to \infty$, thereby confirming a conjecture of Bettin and Drappeau.

Figures (4)

• Figure 1: The function $h(x)$, evaluated at all rationals in $(0,1)$ with denominator at most $80$ (black graph with jumps). For comparison, the plot also shows the function $\frac{\textup{Vol}(4_1)}{2 \pi x} - \frac{3}{2} \log x$ (gray solid line), which is suggested as a continuous approximation to $h(x)$ by Formula \ref{['formula_h']}.
• Figure 2: The function $\psi(x) = h(x) - \textup{Vol}(4_1)/(2 \pi x) + (3/2) \log x$, evaluated at all rationals in $(0,1)$ with denominator at most $80$. Note the apparent self-similar structure of $\psi$. Note also the isolated function values at rationals with small denominators such as $x=1/2$ or $x=1/3$, and that $\lim_{x \to 0} \psi(x)$ appears to be $-\frac{\log 3}{4} \approx -0.275$ and $\lim_{x \to 1} \psi(x)$ appears to be 0, in accordance with the arithmeticity and modularity conjectures.
• Figure 3: The function $\psi(x)$, evaluated at all rationals with denominator at most $600$ in a small neighborhood of $x=1/10$. Note the isolated function value at $x=1/10$, and the very regular behavior when approaching 1/10 from the left or from the right. Note also that the "global" plot in Figure \ref{['fig:psi']} might seem to indicate that $\psi$ consists of a continuous increasing component which is interceded by a discrete decreasing component, and that the values of $\psi$ at rationals are always situated between the corresponding left and right limits, i.e. $\lim_{x \to r^{-}} \psi(x) > \psi(r) > \lim_{x \to r^{+}} \psi(x)$. However, as this figure indicates, this is probably not true for some (small?) rationals, where actually $\lim_{x \to r^{-}} \psi (x) < \psi(r)$, i.e. an initial upward jump is followed by a downward jump. It might still be the case that $\lim_{x \to r^{-}} \psi(x) > \lim_{x \to r^{+}} \psi(x)$ at all rationals $r \in (0,1)$; at least we have not found a counterexample.
• Figure 4: The function $\psi(x)$, evaluated at all rationals with denominator at most $600$ in a small neighborhood of $x=1/\sqrt{2}$. When compared to Figure \ref{['fig:psi_1_10']} above, one can see the different bevahior of $\psi$ near rationals with small denominators and near badly approximable irrationals (note that the scaling is the same in both plots, making them directly comparable).

Theorems & Definitions (28)

• Theorem 1
• Corollary 2
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Proposition 7: Local $5/6$-principle
• Remark