The average number of integral points on the congruent number curves

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-09-11 DOI:10.1016/j.aim.2024.109946

Stephanie Chan

引用次数: 0

Abstract

We show that the total number of non-torsion integral points on the elliptic curves $E_{D} : y^{2} = x^{3} - D^{2} x$ , where D ranges over positive squarefree integers less than N, is $O (N {(\log N)}^{- \frac{1}{4} + ϵ})$ . The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

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全等数曲线上积分点的平均数

我们证明了椭圆曲线 ED:y2=x3-D2x 上的非扭转积分点总数为 O(N(logN)-14+ϵ)，其中 D 的范围是小于 N 的无平方正整数。证明涉及积分二元四元形式的判别降维过程，以及应用希斯-布朗方法估计该族曲线的 2 塞尔默群的平均大小。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.