Explicit limiting proportion of rational points in Frobenian twist families of genus‑1 hyperelliptic curves
Determine the exact value of the limiting proportion, as |d|→∞, of square‑free integers d in the Frobenian family F_b(L) (defined by the local class b∈∏_{v∈Σ}Q_v^×/Q_v^{×2} and total splitting in the splitting field L of the quartic f(x)) for which the quadratic twist C_d: dy^2=f(x) has a rational point, under the assumptions that E is the Jacobian of C with full rational 2‑torsion, the set Σ contains 2, ∞ and all bad reduction and ramified primes of L/Q, that C_d has points everywhere locally, and that the 2^∞‑Selmer rank of E_d/Q is odd. Specifically, prove that the limit equals 1/2 when n_b=1, 1/8 when n_b=2, 5/64 when n_b=3, and 29/1024 when n_b=4, where n_b=dim_F2 S_b is the dimension of the systematic subspace S_b⊂H^1(Q,E[2]) forced into Sel_2(E_d/Q) by the Frobenian conditions.
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Lastly, we formulate a conjecture on the value of this last limit for all possible values of n_b. In the next subsection we will explain our heuristics that lead to this conjecture. Conjecture Let C, E, and Σ be as in Corollary 1.??. Let b∈∏_{v∈Σ}Q_v{×}/Q_v{×2} be such that for all d∈F_b(L), the curve C_d has points everywhere locally and the 2∞‑Selmer rank of E_d/ Q is odd. Let α(r) be as in Theorem 1.??, and n_b=dim S_b as introduced before Theorem 1.??. Then we have [limit equals] 1/2, when n_b=1; 1/8, when n_b=2; 5/64, when n_b=3; 29/1024, when n_b=4.