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

Kin Ming Tsang

doi:10.1016/j.jnt.2025.09.012

paper

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

arXiv:2408.01643 · doi:10.1016/j.jnt.2025.09.012

Abstract

We consider a variant of the strong multiplicity one theorem. Let $I ¨ _{1}$ and $I ¨ _{2}$ be two unitary cuspidal automorphic representations for $GL (2)$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $a_{v} (I ¨ _{1}) > a_{v} (I ¨ _{2})$ , where $a_{v} (I ¨ _{i})$ is the trace of Langlands conjugacy class of $I ¨ _{i}$ 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 $a_{v} (I ¨ _{1}) \neq = a_{v} (I ¨ _{2})$ .

Revised version