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.