Shifted Convolution Sum for $GL(3) \times GL(2)$ with Weighted Average

TL;DR

This paper establishes a non-trivial bound for a weighted average shifted convolution sum involving automorphic forms on $GL(3)$ and $GL(2)$, advancing understanding of their Fourier coefficient correlations.

Contribution

It proves a new bound for the weighted average shifted convolution sum for $GL(3) imes GL(2)$, extending previous results to a broader range of parameters.

Findings

Proves a bound of $X^{1- ext{delta}+ ext{epsilon}}$ for the sum

Validates the bound for $H$ in the range $X^{1/4+ ext{delta}} o X$

Enhances understanding of Fourier coefficient correlations for automorphic forms

Abstract

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{1/4+\delta} \leq H \leq X$ with $\delta >0$, \[ \frac{1}{H}\sum_{h=1}^\infty \lambda_f(h) V\left( \frac{h}{H}\right)\sum_{n=1}^\infty \lambda_{\pi}(1,n) \lambda_g (n+h) W\left( \frac{n}{X} \right)\ll X^{1-\delta+\epsilon} \] where $V,W$ are smooth compactly supported funtions, $\lambda_f(n), \lambda_g(n)$ and $\lambda_{\pi}(1,n)$ are the normalized n-th Fourier coefficients of $SL(2,\mathbb{Z})$ Hecke-Maass cusp forms $f,g$ and $SL(3,\mathbb{Z})$ Hecke-Maass cusp form $\pi$, respectively.