Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect

Abstract

Let $g$ be a fixed primitive Hecke--Maass cusp form, and let $f$ vary with spectral parameter $k_f=K$, at a fixed level and with trivial nebentypus. We study smooth sums of ${\lambda }_f(n){\lambda }_g(n)e(\alpha n^{\beta })$ for fixed real $\alpha \ne 0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be $L=K^{1/3+a}$ for arbitrarily small fixed $a>0$. We show that, for every $\kappa >0$, there is ${\beta }_0>0$ such that the sum is $O(K^{2/3+\kappa }{X}^{1/2})$ for $1\le X\le K^2$ and $0<\beta \le {\beta }_0$. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound $O(K^{423/224+(5/7)\beta +\epsilon })$ at $X=K^2$ for $0<\beta <5/32$.