全等数曲线上积分点的平均数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics Pub Date : 2024-09-11 DOI:10.1016/j.aim.2024.109946

Stephanie Chan

引用次数: 0

摘要

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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The average number of integral points on the congruent number curves

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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来源期刊

Advances in Mathematics 数学-数学

CiteScore

2.80

自引率

5.90%

发文量

497

审稿时长

7.5 months

期刊介绍： Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.