On quantum modular forms of non-zero weights

TL;DR

This paper investigates quantum modular forms of non-zero weights, establishing their regularity, limiting behavior, and distribution of values for various arithmetic functions satisfying quantum modularity relations.

Contribution

It proves the existence of a continuous extension of quantum modular forms with non-zero weights and analyzes the distribution of their values, connecting to several important arithmetic functions.

Findings

Existence of a limiting continuous function $f^*$ for quantum modular forms.

Values of $f(a/q)$ tend to distribute along the graph of $f^*$.

Limiting measure of these values is shown to be diffuse under certain conditions.

Abstract

We study functions $f$ on $\mathbb Q$ which statisfy a ``quantum modularity'' relation of the shape $$ f(x+1)=f(x), \qquad f(x) - |x|^{-k} f(-1/x) = h(x) $$ where $h:\mathbb R_{\neq 0} \to \mathbb C$ is a function satisfying various regularity conditions. We study the case $\Re(k)\neq 0$. We prove the existence of a limiting function $f^*$ which extends continuously $f$ to $\mathbb R$ in some sense. This means in particular that in the $\Re(k)\neq0$ case the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values $\{f(a/q), 1\leq a<q, (a, q)=1\}$, appropriately normalized, tend to equidistribute along the graph of $f^*$, and we prove that under natural hypotheses the limiting measure is diffuse. We apply these results to obtain limiting distributions of values and continuity results for several arithmetic functions known to satisfy the above quantum modularity: higher weight modular symbols associated to holomorphic cusp forms; Eichler integral associated to Maass forms; a function of Kontsevich and Zagier related to the Dedekind $\eta$-function; and generalized cotangent sums.