Moments of the Rank of Elliptic Curves

Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the Lfunction L(ED, s) for every quadratic twist ED 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 ED. We derive from this an upper bound for the density of low-lying zeros of L(ED, s) which is compatible with the random matrix models of Katz and Sarnak. We also show that for any unbounded increasing function f on R, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of ED are less than f(D) for almost all D.