Moments of the rank of elliptic curves

2003/03/28 by Siman Wong, Wong, Siman
Mathematics · #11G05 #11G40 #11M41 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11G40 #msc:11M41

paper · pdf · doi:10.48550/arxiv.math/0303369

arxiv created 2003/03/28 · openalex publication_date 2003/03/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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∈\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.

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