Effective multiplicity one on GLn and narrow zero-free regions for Rankin-Selberg L-functions

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn×GLn′ . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s, π × π̃), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GLn is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor. Let A be the ring of adeles over a number field F and let π and π be two cuspidal representations of GLn(A) with restricted tensor product decompositions π = ⊗vπv and π = ⊗vπ ′ v over all places v of F . The strong multiplicity one theorem asserts that if πv ≃ π v for all but finitely many places v, then π = π . This was proven by Piatetski-Shapiro [PS] using the uniqueness of the Kirillov model and then by Jacquet and Shalika [J-S] using Rankin-Selberg L-functions. Much more can be said however about the extent to which agreement of local factors on a suitable subset of the primes determines global equality. For instance, Moreno has shown [Mo1] that for some finite Y (π, π) the condition that πp ≃ π p for spherical non-archimedean p with absolute norm Np ≤ Y (π, π) is sufficient to imply π = π. From the analytic perspective, the crucial issue are the zeros of Rankin-Selberg L-functions: under GRH for both L(s, π × π̃) and L(s, π × π̃), if the analytic conductors of π and π are less than Q, then Y (π, π) = O(log Q) (see, for example, [G-H]). One wants to give an upper bound on Y (π, π) which grows moderately in Q without assuming a Riemann Hypothesis. In certain settings, this can be done through non-analytic means. As an example, Murty [Mu] used the Riemann-Roch theorem on the modular curve X0(N) to show that when π and π correspond to holomorphic modular forms of level N and even weight k, then Y (π, π) = O(kN log logN). For the case of Maass forms on the upper half plane, Huntley [H] used the method of Rayleigh quotients to show that Y (π, π) grows at most linearly in the eigenvalue. More recently, Baba, Chakraborty, and Petridis [B-C-P] proved a linear bound in the level and weight of holomorphic Hilbert modular forms, again using Rayleigh quotients.