Moments of the Rank of Elliptic Curves

Miller, Steven J. ; Wong, Siman

Canadian journal of mathematics, Tome 64 (2012) no. 1, pp. 151-182

Résumé

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

Détail

DOI : 10.4153/CJM-2011-037-7

Mots-clés : 11G05, 11G40, elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank