Averages of shifted convolution sums for $GL(3)\times GL(2)$

Abstract

Let $A_f(1,n)$ be the normalized Fourier coefficients of a $GL(3)$ Maass cusp form $f$ and let $a_g(n)$ be the normalized Fourier coefficients of a $GL(2)$ cusp form $g$. Let $\hat{I}»(n)$ be either $A_f(1,n)$ or the triple divisor function $d_3(n)$. It is proved that for any $\hat{I}\mu >0$, any integer $r\ge 1$ and $r^{5/2}{X}^{1/4+7\hat{I}´/2}\le H\le X$ with $\hat{I}´>0$, $$\frac{1}{H}\sum _{h\ge 1}W\left(\frac{h}{H}\right)\sum _{n\ge 1}\hat{I}»(n)a_g(rn+h)V\left(\frac{n}{X}\right)\ll X^{1-\hat{I}´+\hat{I}\mu },$$ where $V$ and $W$ are smooth compactly supported functions, and the implied constants depend only on the associated forms and $\hat{I}\mu$.

15 pages. Comments are welcome!