Moments of the rank of elliptic curves

Fix an elliptic curve $E\mathrm{/}\text{Q}$, and assume the generalized Riemann hypothesis for the $L$-function $L(E_D,s)$ for every quadratic twist $E_D$ of $E$ by $D\in \mathbb{Z}$. 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 $\mathbb{R}$, 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.