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

Abstract

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with normalized Fourier coefficients $\hat{I}{»}_f(n)$ and $\hat{I}{»}_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$\sum _{n=1}^{\infty }\hat{I}{»}_f(n)\hat{I}{»}_g(n)e\left(t\ddot{I}†\left(\frac{n}{X}\right)\right)V\left(\frac{n}{X}\right),$$ where $e(x)=e^{2\ddot{I}€ix}$, $V(x)\in {\mathcal{C}}_c^{\infty }(1,2)$, $t\ge 1$ is a large parameter and $\ddot{I}†(x)$ is some nonlinear real valued smooth function. Applications of these estimates include a subconvex bound for the Rankin-Selberg $L$-function $L(s,f\otimes g)$ in the $t$-aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain ${\mathrm{GL}}_5$ Eisenstein series $$\sum _{n\le X}\hat{I}{»}_{1⊞(f\times g)}(n)=L(1,f\times g)X+O(X^{\frac{2}{3}-\frac{1}{356}+\epsilon })$$ for any $\epsilon >0$.

30 pages. Comments welcome! arXiv admin note: text overlap with arXiv:1912.09772