Given integers $m$, $n$ and $k$, we give an explicit formula with an optimal error term (with square root cancelation) for the Petersson trace formula involving the $m$-th and $n$-th Fourier coefficients of an orthonormal basis of $S_k(N)^*$ (the weight $k$ newforms with fixed square-free level $N$) provided that $|4 \pi \sqrt{mn}- k|=o(k^{\frac{1}{3}})$. Moreover, we establish an explicit formula with a power saving error term for the trace of the Hecke operator $\mathcal{T}_n^*$ on $S_k(N)^*$ averaged over $k$ in a short interval. By bounding the second moment of the trace of $\mathcal{T}_{n}$ over a larger interval, we show that the trace of $\mathcal{T}_n$ is unusually large in the range $|4 \pi \sqrt{n}- k| = o(n^{\frac{1}{6}})$. As an application, for any fixed prime $p$ with $\gcd(p,N)=1$, we show that there exists a sequence $\{k_n\}$ of weights such that the error term of Weyl's law for $\mathcal{T}_p$ is unusually large and violates the prediction of arithmetic quantum chaos. In particular, this generalizes the result of Gamburd, Jakobson and Sarnak~\cite[Theorem 1.4]{Gamburd} with an improved exponent.

中文翻译：

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Hecke 算子的渐近迹公式

给定整数 $m$、$n$ 和 $k$，我们为涉及 $m$-th 和 $n$-th 傅立叶系数的 Petersson 迹公式给出了一个带有最佳误差项（具有平方根抵消）的显式公式$S_k(N)^*$ 的标准正交基（权重 $k$ 新形式，具有固定的无平方水平 $N$），条件是 $|4 \pi \sqrt{mn}- k|=o(k^ {\frac{1}{3}})$。此外，我们为Hecke算子$\mathcal{T}_n^*$在$S_k(N)^*$上的迹线建立了一个带有节能误差项的显式公式，在短时间内平均超过$k$。通过在更大的区间上限制 $\mathcal{T}_{n}$ 迹的二阶矩，我们表明 $\mathcal{T}_n$ 的迹在 $|4 \pi 范围内异常大\sqrt{n}- k| = o(n^{\frac{1}{6}})$。作为一个应用程序，对于任何固定素数 $p$ 且 $\gcd(p,N)=1$，我们证明存在一个序列 $\{k_n\}$ 的权重，使得 $\mathcal{T}_p$ 的外尔定律的误差项异常大，并且违反了算术量子混沌的预测。特别是，这概括了 Gamburd、Jakobson 和 Sarnak 的结果~\cite[Theorem 1.4]{Gamburd} 具有改进的指数。