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 \(E/\mathbb{Q}\) be an elliptic curve with one rational 2-torsion point. For quadratic twists \(E^{(d)}\), the 2-Selmer rank \(s_2(E^{(d)})\) satisfies: \(\liminf_{X \to \infty} \frac{\#\{|d| \leq X : s_2(E^{(d)}) \leq 1\}}{\#\{|d| \leq X\}} \geq \frac{1}{2}\). This is a density result, not a precise constant. The 2-torsion structure is \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\) 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 \(\mathbb{Z}^4\) with norm \(\sqrt{2}\). The 2-tors Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com