Rational torsion on optimal curves and rank-one quadratic twists

Abstract: When an elliptic curve of square-free conductor N has a rational point of odd prime order , Dummigan (2005) in explicitly constructed a rational point of order l on the optimal curve E, isogenous over to , 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 such that a positive proportion of quadratic twists of has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) . [ABSTRACT FROM AUTHOR]