Sharp Laws for 2-Selmer Ranks in Quadratic Twists via Fitting Ideals — E8 Intelligence Research

FINDING: The 2-primary Tate-Shafarevich group and Selmer-rank statistics in quadratic twist families exhibit sharp, predictable distributional laws tied to BSD and Cohen-Lenstra heuristics, with explicit control via higher Fitting ideals and parity disparities. | MATH: Let \(E\) be an elliptic curve over \(\mathbb{Q}\), \(E_d\) its quadratic twist by \(d\). The 2-Selmer rank \(s_2(E_d)\) satisfies \(s_2(E_d) \equiv \dim_{\mathbb{F}_2} E(\mathbb{Q}_2)[2] \pmod{2}\) (parity). Smith's theorem: for a positive proportion of \(d\), \(\dim_{\mathbb{F}_2} \mathrm{Sel}_2(E_d) - \dim_{\mathbb{F}_2} E_d(\mathbb{Q})[2] = 0\) or 1, i.e., the 2-primary part of \(\mathrm{III}(E_d)[2^\infty]\) is bounded. Longo's higher Fitting ideals: \(\mathrm{Fitt}_i(\mathrm{III}(A_s))\) control the \(p\)-torsion structure, with \(\mathrm{Fitt}_0(\mathrm{III}) \subseteq (L(1,\chi_d))\) in the anticyclotomic setting. Mazur's disparity: the average 2-Selmer rank in twist families is \(\frac{1}{2}\) (half-integer), re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com