Kolyvagin-inspired conjecture on all 2-Selmer ranks in quadratic twist families

Let ${\mathcal {E}}$ be a family of type (B) or (C). For a $\Sigma$-equivalence class ${\mathfrak{X}}$, let $r({\mathfrak{X}})$ denote its parity parameter and let $\mathbf t_{\mathfrak{X}}=(t_i)$ be its associated parameter. Kolyvagin-inspired conjecture. For $r=0$ and $r=1$, there exists a $\Sigma$-equivalence class ${\mathfrak{X}}$ with $r({\mathfrak{X}})=r$ and

$$\max_i(t_i)\leq r.$$

Consequently, if ${\mathcal {E}}$ is a quadratic twist family of elliptic curves over ${\mathbb {Q}}$ with full rational $2$-torsion points, then for every integer $d\geq0$,

$$\operatorname{Prob}\left(\dim_{{\mathbb {F}}_2}S(E)=d\ \big|\ E\in{\mathcal {E}}\right)>0.$$

In particular, a positive density of curves in ${\mathcal {E}}$ have trivial $2^\infty$-part of their Shafarevich–Tate groups. The conjecture is motivated by the $p=2$ case of a conjecture of Kolyvagin; the supplied text gives no resolution status.