The analytic strong multiplicity one theorem for GLm(AK)

Let π=⊗πv and be two irreducible, automorphic, cuspidal representations of GLm(AK). Using the logarithmic zero-free region of Rankin–Selberg L-function, Moreno established the analytic strong multiplicity one theorem if at least one of them is self-contragredient, i.e. π and π′ will be equal if they have finitely many equivalent local components , for which the norm of places are bounded polynomially by the analytic conductor of these cuspidal representations. Without the assumption of the self-contragredient for π,π′, Brumley generalized this theorem by a different method, which can be seen as an invariant of Rankin–Selberg method. In this paper, influenced by Landau's smooth method of Perron formula, we improved the degree of Brumley's polynomial bound to be 4m+ε.