A shifted convolution sum for $GL(3)\times GL(2)$

TL;DR

This paper provides an improved upper bound for a shifted convolution sum involving Fourier coefficients of $SL(3)$ and $SL(2)$ cusp forms, advancing understanding in automorphic forms and analytic number theory.

Contribution

It establishes a sharper upper bound for the sum, refining previous results and demonstrating progress in estimating shifted convolution sums for higher rank groups.

Findings

Proves an upper bound of $O(X^{21/22+\, ext{epsilon}})$ for the sum.

Improves upon recent bounds by Munshi.

Enhances techniques for analyzing automorphic Fourier coefficients.

Abstract

In this paper, we estimate the shifted convolution sum \[\sum_{n\geqslant1}\lambda_1(1,n)\lambda_2(n+h)V\Big(\frac{n}{X}\Big),\] where $V$ is a smooth function with support in $[1,2]$, $1\leqslant|h|\leqslant X$, $\lambda_1(1,n)$ and $\lambda_2(n)$ are the $n$-th Fourier coefficients of $SL(3,\mathbf{Z})$ and $SL(2,\mathbf{Z})$ Hecke-Maass cusp forms, respectively. We prove an upper bound $O(X^{\frac{21}{22}+\varepsilon})$, updating a recent result of Munshi.