Analytic twists of $\rm GL_3\times \rm GL_2$ automorphic forms

Let $π$ be a Hecke--Maass cusp form for $\rm SL_3(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_π(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$ with normalized Hecke eigenvalues $λ_f(n)$. In this paper, we are concerned with obtaining nontrivial estimates for the sum \begin{equation*} \sum_{r,n\geq 1}λ_π(n,r)λ_f(n)e\left(t\,φ(r^2n/N)\right)V\left(r^2n/N\right), \end{equation*} where $e(x)=e^{2πix}$, $V(x)\in \mathcal{C}_c^{\infty}(0,\infty)$, $t\geq 1$ is a large parameter and $φ(x)$ 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}λ_π(r,n)λ_f(n)\ll_{π, 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.

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