Summing Hecke eigenvalues over polynomials

Abstract

In this paper we estimate sums of the form \(\sum _{n\le X}|a_{{\text {Sym}}^m \pi }(|f(n)|)|\), for symmetric power lifts of automorphic representations \(\pi \) attached to holomorphic forms and polynomials \(f(x)\in {\mathbb {Z}}[x]\) of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when \(\deg (f)\le 4\). Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves.