Average shifted convolution sum for $GL(d_1)\times GL(d_2)$

TL;DR

This paper establishes a power-saving bound for average shifted convolution sums involving Fourier coefficients of Hecke--Maass cusp forms on high-rank groups, extending previous results and approaching the subconvexity threshold.

Contribution

It provides a new nontrivial bound for shifted convolution sums on $GL(d_1) imes GL(d_2)$ with $d_i \\ge 4$, advancing understanding of automorphic forms and $L$-functions.

Findings

Bound holds for shift $H \\ge N^{1 - 4/(d_1+d_2) + \\varepsilon}$

Extends previous results for $d_1 = d_2 + 1$ and $d_1 = d_2$

Reaches the critical threshold for potential subconvexity bounds

Abstract

We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{\pi_1}(n)\, A_{\pi_2}(n+h), $$ where $A_{\pi_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $\pi_i$ for $\mathrm{SL}(d_i,\mathbb{Z})$ with $d_i\ge 4$, $i=1,2$. We establish a nontrivial power-saving bound of $B(H,N)$ for the range of the shift $H\ge N^{1-\frac{4}{d_1+d_2}+\varepsilon}$ for any $\varepsilon>0$. For the cases $d_1 = d_2 + 1$ and $d_1 = d_2$, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when $d_1 = d_2$, we reach the critical threshold $H\ge N^{1-2/d+\varepsilon}$ such that any further improvement in this range yields a subconvexity bound for the corresponding standard $L$-function in the $t$-aspect.