EFFECTIVE MULTIPLICITY ONE ON GLN AND NARROW ZERO-FREE REGIONS FOR RANKIN-SELBERG L-FUNCTIONS By FARRELL BRUMLEY

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. Introduction. 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 Lfunctions. 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 nonarchimedean 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( log2 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 log N). 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-CP] proved a linear bound in the level and weight of holomorphic Hilbert modular forms, again using Rayleigh quotients. Manuscript received December 13, 2004; revised January 4, 2005. American Journal of Mathematics 128 (2006), 0–00. 1