1. Rational torsion on optimal curves and rank-one quadratic twists
- Author
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Byeon, Dongho and Yhee, Donggeon
- Subjects
- *
TORSION theory (Algebra) , *ALGEBRAIC curves , *QUADRATIC fields , *ELLIPTIC curves , *PRIME numbers , *INFINITE groups , *GEOMETRICAL constructions - Abstract
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]
- Published
- 2011
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