Heegner points on elliptic curves with a rational torsion point

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 E ′ 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 r ∈ { 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] .