Counting rational points on elliptic curves with a rational 2-torsion point

IF 0.6 4区 数学 Q3 MATHEMATICS
F. Naccarato
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引用次数: 1

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.
计算具有有理二扭点的椭圆曲线上的有理点
设$E/\mathbb{Q}$是有理数上的椭圆曲线。Bombieri和Zannier的工作表明,如果$E$具有全有理$2$-扭转,则Weil高度为$B$的有理点的数目$N_E(B)$为$\exp\left(O\left)。在本文中,我们利用经由$2$-同构的下降方法将这一结果推广到只有一个非平凡有理$2$-扭点的椭圆曲线。此外,我们利用Petsche的一个结果导出了这些曲线的更强上界$N_{E}(B)=\exp\left(O\left。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Rendiconti Lincei-Matematica e Applicazioni
Rendiconti Lincei-Matematica e Applicazioni MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.30
自引率
0.00%
发文量
27
审稿时长
>12 weeks
期刊介绍: The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.
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