Exponential sums with Fourier coefficients of automorphic forms

Abstract

Let $\pi _1,\pi _2, \ldots , \pi _k$ be cuspidal automorphic representations of $\mathrm{GL}(2)$ or $\mathrm{GL}(3)$ with trivial conductor and trivial central character, and $\pi :=\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k$ be the isobaric representation associated to them. We establish that there exists a positive constant $\vartheta _k <1$ such that for any $\alpha \in {\mathbb {R}}$, any $x \ge 1$ and any positive $\varepsilon$, one has

$$\begin{aligned} \sum _{n \le x} \lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n)\mathrm{e}(\alpha n) \ll _{\pi _j,\varepsilon } x^{\vartheta _k+\varepsilon }, \end{aligned}$$

where the implied constant does not depend on $\alpha$. We also consider some isobaric representations containing the trivial representation. As an application, we consider averages of shifted convolution sums of the type

$$\begin{aligned} \sum _{h=1}^{H}\sum _{n=1}^{X}a(n)\lambda _{\pi _1\boxplus \pi _2\boxplus \cdots \boxplus \pi _k}(n+h), \end{aligned}$$

where a(n) is any arithmetic function.