zbMATH — the first resource for mathematics

Quadratic twists of elliptic curves with small Selmer rank. (English) Zbl 1189.14037
Let $$E$$ be an elliptic curve defined over $$\mathbb Q$$ by the equation $$y^2=x^3+ax+b$$. For each element $$d\in \mathbb Q^*/\mathbb Q^{*2}$$ let us denote by $$E_d$$ the quadratic twist of $$E$$ over $$\mathbb Q$$. It is defined by the equation $$dy^2=x^3+ax+b$$. In case $$E$$ has no nontrivial rational $$2$$-torsion points over $$\mathbb Q$$, the author obtains a distribution result concerning the $$2$$-Selmer groups $$\text{Sel}^{(2)}(E_d)$$ of $$E_d$$ when $$d$$ varies. He proves that the cardinality of the set of the natural integers $$|d|<x$$, such that $$d$$ is square-free and $$\dim \text{Sel}^{(2)}(E_d)\leq 1$$, is $$>>{x\over (\log x)^{\alpha}}$$ for some $$\alpha$$ with $$0<\alpha<1$$. It implies distribution informations about the $$2$$-part of the Tate-Shafarevich group of $$E_d$$ and the rank of $$E_d(\mathbb Q)$$.

MSC:
 14H52 Elliptic curves
Full Text: