On the coefficients of automorphic representations over polynomials

Abstract

Let \(\pi \) be a cuspidal automorphic representation of \(\textrm{GL}_2(\mathbb {A}_\mathbb {Q})\) associated to holomorphic forms with Fourier coefficients \(a_{ \pi }(n)\). Consider an automorphic representation \(\Pi \) which is equivalent to \(\textrm{sym}^m \pi \) or \(\pi \times \textrm{sym}^m \pi \). We establish uniform upper bounds for \(\sum _{n\leqslant X} |a_{\Pi } (|f(n)|)|\), where \(f(x)\in \mathbb {Z}[x]\) is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results.