• Corpus ID: 238856924

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

  title={A conjecture of Zagier and the value distribution of quantum modular forms},
  author={Christoph Aistleitner and Bence Borda},
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 argument. More precisely, when JK,0 denotes the colored Jones polynomial of a knot K, Zagier’s modularity conjecture describes the asymptotics of the quotient JK,0(e )/JK,0(e ) as x→ ∞ along rationals with bounded denominators, where γ ∈ SL(2,Z). This problem is most accessible for the figure-eight… 
1 Citations

Figures from this paper

On the order of magnitude of Sudler products
Given an irrational number $\alpha\in(0,1)$, the Sudler product is defined by $P_N(\alpha) = \prod_{r=1}^{N}2|\sin\pi r\alpha|$. Answering a question of Grepstad, Kaltenbock and Neumuller we prove an


Limit laws for rational continued fractions and value distribution of quantum modular forms
We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in~$(0,1]$ ordered by
Modularity and value distribution of quantum invariants of hyperbolic knots
We obtain an exact modularity relation for the $q$-Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot $K$ essentially reduces to the arithmeticity
Quantum modular forms
for all z ∈ H and all matrices γ = ( a b c d ) ∈ SL(2, Z), where k, the weight of the modular form, is a fixed integer. Of course, there are many variants: one can replace the group SL(2, Z) by a
Quantum knot invariants
This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the
On the theorem of Davenport and generalized Dedekind sums
A symmetrized lattice of $2n$ points in terms of an irrational real number $\alpha$ is considered in the unit square, as in the theorem of Davenport. If $\alpha$ is a quadratic irrational, the square
Growth of the Sudler product of sines at the golden rotation number
Abstract We study the growth at the golden rotation number ω = ( 5 − 1 ) / 2 of the function sequence P n ( ω ) = ∏ r = 1 n | 2 sin ⁡ π r ω | . This sequence has been variously studied elsewhere as a
The Zeckendorf expansion of polynomial sequences
In the first part of the paper we prove that the Zeckendorf sum-ofdigits function sZ(n) and similarly defined functions evaluated on polynomial sequences of positive integers or primes satisfy a
Asymptotic behaviour of the Sudler product of sines for quadratic irrationals
We study the asymptotic behaviour of the sequence of sine products $P_n(\alpha) = \prod_{r=1}^n |2\sin \pi r \alpha|$ for real quadratic irrationals $\alpha$. In particular, we study the subsequence
There is increasing interest in q series with jqj = 1. In analysis of these, an important role is played by the behaviour as n!1 of (q; q)n = (1 q)(1 q):::(1 q): We show, for example, that for almost
Partial sums of the cotangent function
Nous prouvons l'existence de formules de reciprocite pour des sommes de la forme $\sum_{m=1}^{k-1} f(\frac{m}k) \cot(\pi\frac{mh}k)$, ou $f$ est une fonction $C^1$ par morceaux, qui met en evidence