Explicit values for the global solubility probability in terms of the systematic Selmer dimension
Determine that, in the setting of Frobenian twist families F_{b,L} for a fixed genus-1 hyperelliptic curve C with Jacobian E/Q (E with full rational 2-torsion) and odd 2^∞-Selmer rank condition, the limiting proportion of d in F_{b,L} with C_d(Q)≠∅ equals 1/2, 1/8, 5/64, and 29/1024 when the systematic subspace dimension n_b equals 1, 2, 3, and 4 respectively.
References
Conjecture. Let C, E, and \Sigma be as in \Cref{cor:intro_3dim}. Let b\in v{\Q}{\Sigma} be such that for all d\in \cF_b, the curve C_d has points everywhere locally and the 2\infty-Selmer rank of E_d/\Q is odd. Let \alpha(r) be as in Theorem \ref{thm:intro_distr}, and n_b=\dim \cS_b as introduced before Theorem \ref{thm:intro_distr}. Then we have
\lim_{X\to \infty}\frac{#{d\in \cF_b: |d| < X, C_d(\Q)\neq \emptyset}{#{d\in \cF_b: |d| < X}= \begin{cases} 1/2, & n_b=1;\ 1/8, & n_b=2;\ 5/64, & n_b=3;\ 29/1024, & n_b=4. \end{cases}