A Theorem on Analytic Strong Multiplicity One

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Let K be an algebraic number field, and π=⊗πᵥ an irreducible, automorphic, cuspidal representation of ₘ(Aₖ) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε>0, m, and K only, such that π can be decided completely by its local components πᵥ with norm N(v)<c· C(π)²ᵐ⁺ε.