Twist Families of Elliptic Curves: At Least Half Have 2-Selmer Rank ≤ 1 — E8 Intelligence Research

FINDING: Quadratic twist families of elliptic curves exhibit constrained 2-Selmer rank distributions, with at least half of twists having rank ≤ 1 when the curve has a single rational 2-torsion point and no cyclic 4-isogeny over its 2-division field. | MATH: Let \(E/Q\) be an elliptic curve with one rational 2-torsion point. For quadratic twists \(E⁽ᵈ⁾\), the 2-Selmer rank \(s_2(E⁽ᵈ⁾)\) satisfies: \(X → ∞ {#\{|d| ≤ X : s_2(E⁽ᵈ⁾) ≤ 1\}}{#\{|d| ≤ X\}} ≥ 1/2\). This is a density result, not a precise constant. The 2-torsion structure is \(Z/2Z × Z/2Z\) for curves with full 2-torsion, but here only one point is rational. | CONNECTION: The D4 Weyl group (order 192) acts on the 2-torsion points of elliptic curves via Galois representations. The root system \(D_4\) has symmetry group of order 192, and its lattice is the checkerboard lattice \(Z^4\) with norm \(√2\). The 2-tors Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com