Probability that local solubility lifts to global points in Frobenian twist families

Establish that for an elliptic curve E over Q with full rational 2-torsion, a fixed genus-1 hyperelliptic curve C: y^2 = f(x) with Jacobian E, and a Frobenian subfamily F_{b,L} of quadratic twists specified by a class b in the product of local square-classes over a finite set of places Σ (containing 2, ∞, primes of bad reduction of E, and primes ramified in the splitting field L of f), the limiting proportion of squarefree d in F_{b,L} for which C_d(Q) is nonempty equals the sum Σ_{r≥0} α(2r+m_b)/(2^{n_b+2r+m_b}−1), where α(k)=∏_{j≥1}(1+2^{−j})^{−1}∏_{j=1}^{k}2/(2^{j}−1), n_b is the dimension of the systematic 2-Selmer subspace S_b⊂H^1(Q,E[2]) forced by the local and Frobenian conditions, and m_b∈{0,1} is the parity parameter determined by Selmer rank parity for F_{b,L}.

Background

The paper studies quadratic twist families defined by Frobenian conditions F_{b,L}, in which local conditions force a systematic subspace S_b inside every 2-Selmer group. The authors prove a distribution theorem (Theorem 1.5) for 2-Selmer ranks in such families and formulate an equidistribution heuristic for the unique nontrivial element of the image of E_d(Q) in Sel_2(E_d)/δ_d(E_d[2]) when the rank is expected to be 1.

Under this heuristic, they predict an explicit limiting probability for global solubility C_d(Q)≠∅ among those twists that are everywhere locally soluble and have odd 2^∞-Selmer rank. The probability depends on the Selmer distribution α(·), the dimension n_b of S_b, and a parity parameter m_b tied to Selmer rank parity in the family.