Sum formula for the limiting proportion via Selmer ranks in Frobenian twist families

Establish that, for an elliptic curve E/Q with full rational 2‑torsion, a finite set Σ containing 2, ∞, all places of bad reduction, and all places ramified in a given Galois field L/Q, and a class b∈∏_{v∈Σ}Q_v^×/Q_v^{×2} such that for all d∈F_b(L) the class [C]∈H^1(Q,E[2]) lies in Sel_2(E_d/Q) and the 2^∞‑Selmer rank of E_d/Q is odd, the limiting proportion of d∈F_b(L) with C_d(Q)≠∅ equals ∑_{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=dim_F2 S_b is the dimension of the systematic subspace forced into Sel_2(E_d/Q), and m_b∈{0,1} is the parity shift determined in Theorem 1.??.

Background

From the equidistribution heuristic for the unique nontrivial element of δ_d(E_d(Q))/δ_d(E_d[2]) inside Sel_2(E_d/Q)/δ_d(E_d[2]), together with the proven 2‑Selmer rank distribution (Theorem 1.??), the authors predict a closed‑form infinite sum governing the limiting probability that the fixed class [C] becomes trivial in H^1(Q,E[2]) via E_d(Q)/2E_d(Q).

The parameters n_b and m_b capture, respectively, the dimension of the systematic subspace added by the Frobenian local conditions and the Selmer parity shift. The function α(k) is the Cohen–Lenstra style distribution governing even/odd 2‑Selmer ranks in these families.