A Theorem on Analytic Strong Multiplicity One

TL;DR

This paper proves a theorem that allows the complete determination of a cuspidal automorphic representation of GL_m over a number field from its local components at places with norm below a certain bound related to its analytic conductor.

Contribution

It establishes an explicit bound on the local components needed to determine an automorphic representation, advancing the understanding of multiplicity one phenomena.

Findings

Provides a bound depending on the analytic conductor and degree m

Enables identification of representations from finitely many local components

Strengthens the analytic strong multiplicity one theorem

Abstract

Let $K$ be an algebraic number field, and $\pi=\otimes\pi_{v}$ an irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$ with analytic conductor $C(\pi)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $\pi$ can be decided completely by its local components $\pi_{v}$ with norm $N(v)<c\cdot C(\pi)^{2m+\varepsilon}.$