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 \\in \\mathcal{F}(X) : s_2(E^d) \\equiv r \\pmod{2}\\} \\geq \\frac{1}{2} \\#\\mathcal{F}(X) + o(X)\\), where \\(\\mathcal{F}(X)\\) is the set of squarefree quadratic twists with conductor \\(\\leq X\\). The parity of \\(s_2(E^d)\\) is governed by the root number \\(w(E^d) = w(E) \\cdot \\chi_d(-N_E)\\), with \\(\\chi_d\\) the quadratic character. The result uses induction on the number of prime factors of \\(d\\), building Selmer groups via exact sequences \\(0 \\to E[2] \\to E[2]^d \\to \\hat{E}[2] \\to 0\\). | CONNECTION: The parity class density \\(\\frac{1}{2}\\) is the binary split — a fundamental symmetry breaking. The root number \\(w(E^d) = \\pm 1\\) corres Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com