Shifted convolution sums for GL(3)×GL(2)

Abstract

For the shifted convolution sum

$$D_h\left(X\right)={\sum }_{m=1}^{\infty }{\lambda }_1(1,m){\lambda }_2(m+h)V\left(\frac{m}{X}\right),$$

where ${\lambda }_1(1,m)$ are the Fourier coefficients of an $SL(3,\mathbb{Z})$ Maass form ${\pi }_1$, ${\lambda }_2\left(m\right)$ are those of an $SL(2,\mathbb{Z})$ Maass or holomorphic form ${\pi }_2$, and $1\le \left|h\right|\ll X^{1+\epsilon }$, we establish the bound

$$D_h\left(X\right){\ll }_{{\pi }_1,{\pi }_2,\epsilon }{X}^{1-1/20+\epsilon }.$$

The bound is uniform with respect to the shift $h$.