Quadratic Twists of Elliptic Curves with Small Selmer Rank

Details Quadratic Twists of Elliptic Curves with Small Selmer Rank Quadratic Twists of Elliptic Curves with Small Selmer Rank

2008-09-29

arxiv.org/abs/0809.5019v3

Mathematics

Number Theory

cited the work of Mazur and Rubin [arxiv:0904.3709]; accepted by Acta Arithmetica

Scientific paper

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.

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