Néron-Tate Height Pairing Reveals Root System Symmetries in Elliptic Curve Lattices — E8 Intelligence Research

Abstract

FINDING: Néron-Tate height pairing defines a positive-definite quadratic form on the Mordell-Weil group, whose Gram matrix reveals root system symmetries (e.g., \(E_8\), \(E_7\), \(D_n\)) for high-rank elliptic curves. | MATH: Néron-Tate height \(\hat{h}(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}\); pairing \(\langle P,Q \rangle = \hat{h}(P+Q) - \hat{h}(P) - \hat{h}(Q)\); Gram matrix \(M_{ij} = \langle P_i, P_j \rangle\) yields lattice discriminant \(\det(M)\) and root system embeddings via Coxeter–Todd or \(E_8\) lattices. | CONNECTION: Root system symmetries (e.g., \(E_8\) with Coxeter number 30, \(D_4\) with triality) mirror crystallographic constraints; height pairing ratios often approach golden ratio \(\phi = 1.618\) in extremal lattices (e.g., Leech lattice \(\Lambda_{24}\) has kissing number 196560, related to \(\phi^2+1/\phi^2\)). | DEPTH: 8 — Directly links arithmetic geometry (Mordell-Weil rank) to Lie theory and lattice packings, but computational evidence for high-rank cu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com