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Moments of the Rank of Elliptic Curves

  Published:2011-06-25
 Printed: Feb 2012
  • Steven J. Miller,
    Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, U.S.A.
  • Siman Wong,
    Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, U.S.A.
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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 elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank
MSC Classifications: 11G05, 11G40 show english descriptions Elliptic curves over global fields [See also 14H52]
$L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
11G05 - Elliptic curves over global fields [See also 14H52]
11G40 - $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
 

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