Average of Dirichlet Coefficients of Cuspidal Representations Related to $\GL(2)$

TL;DR

This paper establishes bounds for the average absolute values of Dirichlet coefficients of cuspidal representations on GL(2), extending known estimates to Maass forms without relying on the Ramanujan-Petersson conjecture.

Contribution

It provides new bounds for Dirichlet coefficients of cuspidal representations on GL(2), including symmetric powers, applicable to Maass forms.

Findings

Derived nontrivial bounds for average Dirichlet coefficients

Extended estimates to Maass forms without Ramanujan-Petersson conjecture

Provided bounds for symmetric power representations

Abstract

Let $\pi$ be a cuspidal representation on $\GL(2,\mathbb{A}_{\mathbb{Q}}).$ We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to $\pi;$ and nontrivial upper bound in the case of $\Sym^k\pi,$ $k=2, 3.$ These bounds generalize the known estimates in holomorphic case to Maass forms, without assuming Ramanujan-Petersson conjecture.