Asymptotics for cuspidal representations by functoriality from GL(2)

TL;DR

This paper establishes asymptotic formulas for sums of Fourier coefficients of cuspidal automorphic representations of GL(2), illustrating the influence of functorial liftings and their connection to the distribution of these coefficients.

Contribution

It provides new asymptotic expansions for sums of Fourier coefficients derived from known functorial liftings, highlighting their role in understanding coefficient distribution.

Findings

Asymptotic formulas for sums of Fourier coefficients are derived.

Results reflect the impact of functorial liftings on coefficient distribution.

The work connects functoriality with the value distribution of Fourier coefficients.

Abstract

Let $\pi$ be a unitary automorphic cuspidal representation of $GL_2(\mathbb{Q}_\mathbb{A})$ with Fourier coefficients $\lambda_\pi(n)$. Asymptotic expansions of certain sums of $\lambda_\pi(n)$ are proved using known functorial liftings from $GL_2$, including symmetric powers, isobaric sums, exterior square from $GL_4$ 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.