In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators.

Abstract. We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular those obtained by base change from GL2(Q) to GL2(K) for K a cyclic field of low degree.

We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ (k) (f, 1/2) = 0 is ≥ p k + o(1) with p k > 0 and p k = 1/2 + O(k −2), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result of Conrey on the zeros of high derivatives of Riemann's ξ function lying on the critical line. As an application we obtain new strong unconditional bounds for the average order of vanishing of the forms f (i.e. the analytic rank of the Jacobian variety J 0 (q)).

Using the mollification method, we show that, for large q, at least 2Â3&O(k &2 ) of the set [4 (k) (/, 1Â2)] is non-zero, where 4(/, s) is the complete Dirichlet L-function and / runs over all even primitive characters modulo q. Academic Press

Contents 1. Introduction. Given a large prime N, consider the Jacobian variety J 0 (N) of the modular curve X 0 (N ) = H/ 0 (N). There has been considerable study recently of the rank over Q of these varieties and their quotients, much of it dealing with two questions:• Are there large quotients of J 0 (N) with rank zero over the rationals?• How large a rank can a quotient of J 0 (N) have? See Mazur [9] and Merel [10] for arithmetic approaches to the first question; one consequence of the latter is the existence of a quotient with rank zero and dimension at least N 1/8 (out of a maximum possible dimension of D N = N/12 + ᏻ(1), since N is prime). The progress of Kolyvagin [8] and Gross and Zagier [6] toward the Birch-Swinnerton-Dyer conjecture allows one to investigate these questions through the "analytic rank" of J 0 (N ). This is defined (see, for example, [1]) to be the order of the zero at s = 1 of L(J 0 /Q, s) = f L f (s), where the product is taken over L-functions associated with the Hecke eigenbasis of S 2 ( 0 (N)), the weight-2 holomorphic forms on X 0 (N ) (of which there are D N ). In particular, if one can show that many of the L f (s)'s have order zero or one, then the corresponding quotients provide good answers to each of the questions. Along these lines, Duke [4] has used first and second moments of L-functions to show that at least N/(log N) 2 are nonzero at s = 1, which along with Kolyvagin's results may be used to give a corresponding improvement in the answer to the first question.One can also use Riemann's explicit formula for the L f 's (see Mestre [11]) to control the average order at s = 1, although this approach requires (at least at present) an appeal to the Riemann hypothesis to obtain the desired bounds. In recent papers, Brumer [1] and Murty [12] have used these techniques to show that (assuming the