Moments of the Rank of Elliptic Curves

Abstract

Abstract. Fix an elliptic curve E/Q, and assume the generalized Riemann hypothesis for the L-function L(E D , s) for every quadratic twist E D of E by D ∈ Z. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of E D . It follows from this that, for any unbounded increasing function f on R, the analytic rank and (assuming in addition the Birch-Swinnerton-Dyer conjecture) the number of integral points of E D are less than f (D) for almost all D. We also derive an upper bound for the density of low-lying zeros of L(E D , s) which is compatible with the random matrix models of Katz and Sarnak.