Rational Points of Elliptic Curve y2 = x3＋ k3

作     者：Xia Wu Yan Qin 

作者机构：School of Mathematics Southeast University Nanjing 210096 China Nanjing University Business School Nanjing 210093 China 

出 版 物：《Algebra Colloquium》 (代数集刊（英文版）)

年 卷 期：2018年第25卷第1期

页      面：133-138页

核心收录：

学科分类：07[理学] 081302[工学-建筑设计及其理论] 08[工学] 0813[工学-建筑学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Supported by the Fundamental Research Funds for the Central Universities 国家自然科学基金 the Basic Research Foundation (Natural Science) of Jiangsu Province 

主　　题：elliptic curve rational point class number Pell equation 

摘      要：Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed（Q） is zero. Let E（k） denote the elliptic curve y2 = x3 ＋ k. Then the quadratic twist E（1）d of E（1） by d is the elliptic curve E（d3）： y2 = x3＋ k3. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d = r （rood 24） such that rankE（-d3）（Q） = 0, using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E（k3）（Q） has rank zero.