Analytic twists of $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ automorphic forms

Abstract

Let $\ddot{I}€$ be a Hecke--Maass cusp form for $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$ with normalized Hecke eigenvalues $\hat{I}{»}_{\ddot{I}}€(n,r)$. Let $f$ be a holomorphic or Maass cusp form for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with normalized Hecke eigenvalues $\hat{I}{»}_f(n)$. 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 $e(x)=e^{2\ddot{I}€ix}$, $V(x)\in {\mathcal{C}}_c^{\infty }(0,\infty )$, $t\ge 1$ is a large parameter and $\ddot{I}†(x)$ is some real-valued smooth function. As applications, we give an improved subconvexity bound for $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ $L$-functions in the $t$-aspect, and under the Ramanujan--Petersson conjecture we derive the following bound for sums of $\mathrm{G}{\mathrm{L}}_3\times \mathrm{G}{\mathrm{L}}_2$ 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 $\epsilon >0$, which breaks for the first time the barrier $O(x^{5/7+\epsilon })$ in a work by Friedlander--Iwaniec.

To appear in International Mathematics Research Notices