Double square moments and bounds for resonance sums of cusp forms

Abstract

Let $f$ and $g$ be holomorphic cusp forms for the modular group $S{L}_2(\mathbb{Z})$ of weight $k_1$ and $k_2$ with Fourier coefficients $\hat{I}{»}_f(n)$ and $\hat{I}{»}_g(n)$, respectively. For real $\hat{I}\pm \ne 0$ and $0<{\hat{I}}^2\le 1$, consider a smooth resonance sum $S_X(f,g;\hat{I}\pm ,{\hat{I}}^2)$ of $\hat{I}{»}_f(n)\hat{I}{»}_g(n)$ against $e(\hat{I}\pm n^2)$ over $X\le n\le 2X$. Double square moments of $S_X(f,g;\hat{I}\pm ,{\hat{I}}^2)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking out a small exceptional set of $f$ and $g$, bounds for individual $S_X(f,g;\hat{I}\pm ,{\hat{I}}^2)$ will then be proved. These individual bounds break the resonance barrier of $X^{\frac{5}{8}}$ for $\frac{1}{6}<{\hat{I}}^2<1$ and achieve a square-root cancellation for $\frac{1}{3}<{\hat{I}}^2<1$ for almost all $f$ and $g$ as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.

16 pages