Moments of the rank of elliptic curves
type: Item
instance of: scholarly article
zbMATH Open document ID: 1241.11062
DOI: 10.4153/CJM-2011-037-7
author: Steven J. Miller
author: Siman Wong
published in: Canadian Journal of Mathematics
publication date: 2012-04-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0303369
full work available at URL: CJM-2011-037-7
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 6023030
zbMATH Keywords: elliptic curves
zbMATH Keywords: rank of elliptic curves
zbMATH Keywords: quadratic twists
zbMATH Keywords: low-lying zeros
zbMATH Keywords: integral points
zbMATH Keywords: explicit Formula
MaRDI profile type: Publication
arXiv ID: math/0303369
OpenAlex ID: W2964080107
title: Moments of the rank of elliptic curves
review text: Understanding the rank of rational elliptic curves is a major problem of number theory. Many authors have given upper bounds for the average rank, under various hypotheses. The most striking result so far is the recent result of Bhargava and Shankar [\textit{M. Bhargava} and \textit{A. Shankar}, ``Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank \(0\).'' Preprint: \url{arXiv:1007.0052v1} [math.NT]], who showed unconditionally that the average rank in the family of all elliptic curves is at most \(\frac 76\), by bounding the average order of Selmer groups.NEWLINENEWLINEPrevious bounds on the average rank were obtained under GRH and the Birch and Swinnerton-Dyer Conjecture, which predicts the rank of \(E\) to be equal to its \textit{analytic rank}, that is the order of vanishing of \(L(E,s)\) at \(s=1\). Various authors have studied the average analytic rank in families [\textit{D. Goldfeld}, ``Conjectures on elliptic curves over quadratic fields'', Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 108--118 (1979; Zbl 0417.14031)]; [\textit{A. Brumer}, ``The average rank of elliptic curves. I'', Invent. Math. 109, No. 3, 445--472 (1992; Zbl 0783.14019)]; [\textit{D. R. Heath-Brown}, ``The average analytic rank of elliptic curves'', Duke Math. J. 122, No. 3, 591--623 (2004; Zbl 1063.11013)]; [\textit{M. P. Young}, ``Low-lying zeros of families of elliptic curves'', J. Am. Math. Soc. 19, No. 1, 205-250 (2006; Zbl 1086.11032)]. Heath-Brown proved that in the family of all elliptic curves the average rank is bounded by \(2\), and in the family of quadratic twists it is bounded by \(3/2\). The computation of moments of the rank is implicit in his work, and the goal of the paper under review is to make explicit statements about such moments in the family of quadratic twists of a fixed curve.NEWLINENEWLINEMore precisely it is shown that for a fixed elliptic curve \(E:y^2=x^3+ax+b\) and …
reviewed by: Daniel Fiorilli
description: scientific article; zbMATH DE number 6023030