Shifted convolution sums of $GL_3$ cusp forms with $\theta$-series

TL;DR

This paper establishes an upper bound for shifted convolution sums involving Fourier coefficients of $GL_3$ cusp forms and the representation function of sums of three squares, uniformly over shifts, advancing understanding of automorphic forms and their correlations.

Contribution

It provides a new uniform bound for shifted convolution sums of $GL_3$ cusp forms with the three squares representation function, extending previous results to include shifts.

Findings

Bound for shifted convolution sums with $GL_3$ cusp forms

Uniform estimate over shifts $h$

Improved understanding of automorphic form correlations

Abstract

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $SL_3(\mathbb{Z})$ and $$ r_3(n)=\#\left\{(n_1,n_2,n_3)\in \mathbb{Z}^3:n_1^2+n_2^2+n_3^2=n\right\}. $$ Let $1\leq h\leq X$ and $\phi(x)$ be a smooth function compactly supported on $[1/2,1]$. It is shown that $$ \sum_{n\geq 1}A_f(1,n+h)r_3(n)\phi\left(\frac{n}{X}\right) \ll_{f,\varepsilon} X^{\frac{3}{2}-\frac{1}{8}+\varepsilon} $$ uniformly with respect to the shift $h$.