具有一个有理参数的椭圆曲线族中的秩零二次扭转 \(\mathbb {Q}\)

IF 0.4 Q3 MATHEMATICS

Annales Mathematiques du Quebec Pub Date : 2025-09-12 DOI:10.1007/s40316-025-00253-y

Abhishek Juyal, Mansi Tyagi

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引用次数: 0

摘要

在本文中，我们构造了一组椭圆曲线\(E^A: y^2 = (x + A)(x^2 + A^2)\)，在\(\mathbb {Q}\)上有一个\(2\) -扭转点，并证明了存在无穷多个无平方整数\( d \)，使得\( E^A \) × \( d \)的二次扭转的秩为零。本文推广了M. Xiong的结果：[关于椭圆曲线在\({\mathbb {Q}}\)上的秩零扭转的正比例，数学学报，98:281-288,(2015)]。

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Rank-zero quadratic twists in families of elliptic curves with one rational parameter over \(\mathbb {Q}\)

In this article, we construct a family of elliptic curves \(E^A: y^2 = (x + A)(x^2 + A^2)\) with one \(2\)-torsion point over \(\mathbb {Q}\) and prove that there exist infinitely many square-free integers \( d \) such that the rank of the quadratic twists of \( E^A \) by \( d \) is zero. This work is a generalization of the result of M. Xiong: [On positive proportion of rank-zero twists of elliptic curves over \({\mathbb {Q}}\), J Aust Math Soc 98:281–288, (2015)].

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来源期刊

Annales Mathematiques du Quebec MATHEMATICS-

CiteScore

1.10

自引率

0.00%

发文量

期刊介绍： The goal of the Annales mathématiques du Québec (formerly: Annales des sciences mathématiques du Québec) is to be a high level journal publishing articles in all areas of pure mathematics, and sometimes in related fields such as applied mathematics, mathematical physics and computer science. Papers written in French or English may be submitted to one of the editors, and each published paper will appear with a short abstract in both languages. History: The journal was founded in 1977 as „Annales des sciences mathématiques du Québec”, in 2013 it became a Springer journal under the name of “Annales mathématiques du Québec”. From 1977 to 2018, the editors-in-chief have respectively been S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea. Les Annales mathématiques du Québec (anciennement, les Annales des sciences mathématiques du Québec) se veulent un journal de haut calibre publiant des travaux dans toutes les sphères des mathématiques pures, et parfois dans des domaines connexes tels les mathématiques appliquées, la physique mathématique et l''informatique. On peut soumettre ses articles en français ou en anglais à l''éditeur de son choix, et les articles acceptés seront publiés avec un résumé court dans les deux langues. Histoire: La revue québécoise “Annales des sciences mathématiques du Québec” était fondée en 1977 et est devenue en 2013 une revue de Springer sous le nom Annales mathématiques du Québec. De 1977 à 2018, les éditeurs en chef ont respectivement été S. Dubuc, R. Cléroux, G. Labelle, I. Assem, C. Levesque, D. Jakobson, O. Cornea.