Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415663 | Journal of Number Theory | 2012 | 8 Pages |
Abstract
Using Heegner points on elliptic curves, we construct points of infinite order on certain elliptic curves with a Q-rational torsion point of odd order. As an application of this construction, we show that for any elliptic curve E defined over Q which is isogenous to an elliptic curve Eâ² defined over Q of square-free conductor N with a Q-rational 3-torsion point, a positive proportion of quadratic twists of E have (analytic) rank r, where râ{0,1}. This assertion is predicted to be true unconditionally for any elliptic curve E defined over Q due to Goldfeld (1979) [Go], but previously has been confirmed unconditionally for only one elliptic curve due to Vatsal (1998) [V1].
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Dongho Byeon,