Asymptotics for cuspidal representations by functoriality from GL(2)

Abstract

Let $\pi$ be a unitary automorphic cuspidal representation of $G{L}_2({\mathbb{Q}}_{\mathbb{A}})$ with Fourier coefficients ${\lambda }_{\pi }(n)$. Asymptotic expansions of certain sums of ${\lambda }_{\pi }(n)$ are proved using known functorial liftings from $G{L}_2$, including symmetric powers, isobaric sums, exterior square from $G{L}_4$ and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of ${\lambda }_{\pi }(n)$ on integers, squares, cubes and fourth powers.