Parity Distribution of 2-Selmer Ranks in Quadratic Twists — E8 Intelligence Research

FINDING: Distribution of 2-Selmer ranks in quadratic twist families of elliptic curves; at least half of twists have 2-Selmer rank equal to a fixed parity class, with partial 2-torsion conditions. | MATH: For an elliptic curve \(E/K\) with a single rational 2-torsion point and no cyclic 4-isogeny over \(K(E[2])\), the 2-Selmer rank \(s_2(E^d)\) satisfies: \(#\{d ∈ F(X) : s_2(E^d) ≡ r {2}\} ≥ 1/2 #F(X) + o(X)\), where \(F(X)\) is the set of squarefree quadratic twists with conductor \(≤ X\). The parity of \(s_2(E^d)\) is governed by the root number \(w(E^d) = w(E) · χ_d(-N_E)\), with \(χ_d\) the quadratic character. The result uses induction on the number of prime factors of \(d\), building Selmer groups via exact sequences \(0 → E[2] → E[2]^d → Ê[2] → 0\). | CONNECTION: The parity class density \(1/2\) is the binary split — a fundamental symmetry breaking. The root number \(w(E^d) = ± 1\) corres Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com