# A conjectural refinement of strong multiplicity one for GL(n)

## TL;DR

This paper explores how conjectures on automorphy and cuspidality influence the size of the set of places where two automorphic representations differ, providing new bounds and results especially for GL(3).

## Contribution

It proposes a conjectural refinement of strong multiplicity one for GL(n) based on automorphy and cuspidality conjectures, offering new lower bounds on the difference set.

## Key findings

- Lower bounds on the size of the set S where automorphic representations differ.
- Results specific to GL(3) representations.
- Implications for automorphy and cuspidality conjectures.

## Abstract

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this note, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin-Selberg products imply lower bounds on the size of S. We also obtain further results for GL(3).