Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)

TL;DR

This paper establishes a lower bound on the density of places where two non-twist-equivalent automorphic representations for GL(2) have larger trace coefficients, improving bounds on their differences.

Contribution

It provides a new lower bound on the density of places with larger trace values for pairs of automorphic representations, refining previous bounds.

Findings

Lower bound for the density of places with larger trace coefficients.

Improved bounds on the density of places where traces differ.

Enhanced understanding of automorphic representation comparisons.

Abstract

We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$ and $\pi_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1}) \right\rvert > \left\lvert a_{v}(\pi_{2}) \right\rvert$, where $a_{v}(\pi_{i})$ is the trace of Langlands conjugacy class of $\pi_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1})\right\rvert \neq \left\lvert a_{v}(\pi_{2}) \right\rvert$.