Let $\pi$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{ Z})$ with normalized Hecke eigenvalues $\lambda_{ \pi}(n,r)$. 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 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--Petersson 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_{ \pi, f, \varepsilon} 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 by Friedlander--Iwaniec.
Categories: math.NT