A Theorem on Analytic Strong Multiplicity One

Let $K$ be an algebraic number field, and $\pi =\otimes {\pi }_v$ an irreducible, automorphic, cuspidal representation of ${\text{GL}}_m({\mathbb{A}}_K)$ 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 $\epsilon >0,m,$ and $K$ only, such that $\pi$ can be decided completely by its local components ${\pi }_v$ with norm $N(v)<c\cdot C(\pi)^{2m+\varepsilon}.$