Rational torsion on optimal curves and rank-one quadratic twists

JOURNAL OF NUMBER THEORY

卷:131

Article

Byeon, Dongho¹  Yhee, Donggeon¹ 

[1] Seoul Natl Univ, Dept Math, Seoul, South Korea

关键词: Rank;    Torsion;    Elliptic curves;    Twists;   

DOI  :  10.1016/j.jnt.2010.10.005

来源: Elsevier

【 摘 要 】

When an elliptic curve E'/Q of square-free conductor N has a rational point of odd prime order I inverted iota N, Dummigan (2005) in [Du] explicitly constructed a rational point of order I on the optimal curve E, isogenous over Q to E', under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves E'/Q such that a positive proportion of quadratic twists of E' has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) [B-J-K]. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free