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Moments of the Rank of Elliptic Curves |
Canad. J. Math. 64(2012), 151-182
Published:2011-06-25
Printed: Feb 2012
Printed: Feb 2012
- Steven J. Miller,
- Siman Wong,
Abstract
Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. 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 $E_D$. We
derive from this an upper bound for the density of low-lying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
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 $D$.
| Keywords: | elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank |
| MSC Classifications: |
11G05 - Elliptic curves over global fields [See also 14H52] 11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] |