Let $\pi$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda_{\pi}(n,r)$ and let $f$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$ with normalized Hecke eigenvalues $\lambda_f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum \begin{equation*} \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left(t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right), \end{equation*} where $e(x):=e^{2\pi ix}$, $V(x)\in \mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ is a large parameter and $\varphi(x)$ is some nonlinear real-valued smooth function. As applications, we give an improved subconvexity bound for $\rm GL_3\times \rm GL_2$ $L$-functions in the $t$-aspect, and under the Ramanujan conjecture we derive the following bound for sums of $\rm GL_3\times \rm GL_2$ Fourier coefficients \begin{equation*} \sum_{r^2n\leq x}\lambda_{\pi}(r,n)\lambda_f(n)\ll x^{5/7-1/364+\varepsilon} \end{equation*} for any $\varepsilon>0$, which breaks for the first time the barrier $O(x^{5/7+\varepsilon})$ in a work of Friedlander--Iwaniec.

中文翻译：

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GL3 × GL2 自守形式的解析扭曲

令 $\pi$ 是具有归一化 Hecke 特征值 $\lambda_{\pi}(n,r)$ 的 $\rm SL_3(\mathbb{Z})$ 的 Hecke--Maass 尖点形式，并令 $f$ 为$\rm SL_2(\mathbb{Z})$ 的全纯或 Maass 尖点形式，具有归一化的 Hecke 特征值 $\lambda_f(n)$。在本文中，我们关注获得总和 \begin{equation*} \sum_{r,n\geq 1}\lambda_{\pi}(n,r)\lambda_f(n)e\left( t\,\varphi(r^2n/N)\right)V\left(r^2n/N\right),\end{equation*} 其中 $e(x):=e^{2\pi ix} $, $V(x)\in \mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ 是一个大参数，$\varphi(x)$ 是一些非线性实数-值平滑函数。作为应用，我们在 $t$ 方面为 $\rm GL_3\times \rm GL_2$ $L$-functions 提供了改进的子凸边界，