Shifted convolution sums for $GL(3)\times GL(2)$

TL;DR

This paper establishes a uniform bound for shifted convolution sums involving Fourier coefficients of $SL(3)$ and $SL(2)$ Maass forms, advancing understanding of their analytic behavior in number theory.

Contribution

It provides a new uniform bound for shifted convolution sums of $GL(3) imes GL(2)$ Fourier coefficients, improving previous estimates in the field.

Findings

Bound $D_h(X) o X^{1 - 1/20 + ext{small}}$ for large $X$

Uniformity of the bound with respect to the shift $h$

Application to analytic properties of automorphic forms

Abstract

For the shifted convolution sum $$ D_h(X)=\sum_{m=1}^\infty\lambda_1(1,m)\lambda_2(m+h)V(\frac{m}{X}) $$ where $\lambda_1(1,m)$ are the Fourier coefficients of a $SL(3,\mathbb Z)$ Maass form $\pi_1$, and $\lambda_2(m)$ are those of a $SL(2,\mathbb Z)$ Maass or holomorphic form $\pi_2$, and $1\leq |h| \ll X^{1+\varepsilon}$, we establish the bound $$ D_h(X)\ll_{\pi_1,\pi_2,\varepsilon} X^{1-(1/20)+\varepsilon}. $$ The bound is uniform with respect to the shift $h$.