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 \(E_d(\mathbb Q)\).

14H52 Elliptic curves
Full Text: DOI arXiv