Double square moments and bounds for resonance sums of cusp forms

Abstract

Let f and g be holomorphic cusp forms for the modular group SL2(Z) of weight k1 and k2 with Fourier coefficients λf(n) and λg(n), respectively. For real α≠0 and 0<β≤1, consider a smooth resonance sum SX(f,g;α,β) of λf(n)λg(n) against e(αnβ) over X≤n≤2X. Double square moments of SX(f,g;α,β) over both f and g are nontrivially bounded when their weights k1 and k2 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 SX(f,g;α,β) shall then be proved. These individual bounds break the resonance barrier of X5/8 for 1/6<β<1 and achieve a square-root cancellation for 1/3<β<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.