Quadratic Twists of Elliptic Curves with Small Selmer Rank

@article{Chang2008QuadraticTO,
  title={Quadratic Twists of Elliptic Curves with Small Selmer Rank},
  author={Sungkon Chang},
  journal={arXiv: Number Theory},
  year={2008}
}
Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial. 
1 Citations
Ranks of twists of elliptic curves and Hilbert’s tenth problem
In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it hasExpand

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