Double square moments and bounds for resonance sums of cusp forms

TL;DR

This paper establishes new bounds for resonance sums of cusp form Fourier coefficients, demonstrating square-root cancellation and breaking previous resonance barriers, providing evidence for Hypothesis S.

Contribution

It introduces a novel analysis of double square moments of resonance sums for cusp forms, connecting them to automorphic forms on GL(4) and improving bounds for individual sums.

Findings

Bounds for double moments are nontrivially established as weights grow.

Individual resonance sums break the resonance barrier for certain parameters.

Results support Hypothesis S for cusp forms over integers.

Abstract

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ 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 $S_X(f,g;\alpha,\beta)$ will then be proved. These individual bounds break the resonance barrier of $X^\frac58$ for $\frac16<\beta<1$ and achieve a square-root cancellation for $\frac13<\beta<1$ 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.