Moments of the rank of elliptic curves
| dc.creator | Wong, Siman | |
| dc.date | 2003-03-28 | |
| dc.date.accessioned | 2026-07-07T04:56:29Z | |
| dc.date.available | 2026-07-07T04:56:29Z | |
| dc.description | 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 $D\in\Z$. 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. | |
| dc.identifier | https://arxiv.org/abs/math/0303369 | |
| dc.identifier | http://arxiv.org/abs/math/0303369 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66935 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05; 11M41, 11G40 | |
| dc.title | Moments of the rank of elliptic curves | |
| dc.type | text |