Moments of the rank of elliptic curves

Original · EN

Fix an elliptic curve E/, and assume the generalized Riemann hypothesis for the L-function L(ED, s) for every quadratic twist ED of E by D∈. 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. It follows from this that, for any unbounded increasing function f on, the analytic rank and (assuming in addition the Birch-Swinnerton-Dyer conjecture) the number of integral points of ED are less than f(D) for almost all D. We also derive 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.