论文标题
非零重量的量子模块化形式
On quantum modular forms of non-zero weights
论文作者
论文摘要
我们在$ \ mathbb q $上研究功能$ f $ $$ f（x+1）= f（x），\ qquad f（x） - | x |^{ - k} f（-1/x）= h（x）$$ 其中$ h：\ mathbb r _ {\ neq 0} \ to \ mathbb c $是满足各种规律性条件的功能。我们研究案例$ \ re（k）\ neq 0 $。 我们证明存在限制函数$ f^*$，从某种意义上说，它将$ f $连续扩展到$ \ mathbb r $。这特别意味着在$ \ re（k）\ neq0 $案例中，量子模块化表格本身必须至少具有一定级别的规律性。 我们推断出值$ \ {f（a/q），1 \ leq a <q，（a，q）= 1 \} $，适当地归一化，倾向于沿$ f^*$的图均衡，我们证明，在自然假设下，有限的措施是扩散的。 我们应用这些结果来获得几种算术函数的限制值和连续性结果的分布，已知以满足上述量子模块：较高的重量模块化符号与全体形态尖端相关；艾希勒（Eichler）与玛斯形式相关的积分； Kontsevich和Zagier的功能与Dedekind $η$ function相关；和广义的cotangent总和。
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 $η$-function; and generalized cotangent sums.
