Mordell-Weil Lattice Realizes Root Systems from Elliptic Curves — E8 Intelligence Research

FINDING: Mordell-Weil theorem establishes that rational points on an elliptic curve form a finitely generated abelian group, linking arithmetic to lattice geometry via the Mordell-Weil lattice. | MATH: \( E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}} \), where \( r \) is the rank; Mordell-Weil lattice is a positive-definite lattice with inner product given by the Néron-Tate height pairing. | CONNECTION: The Mordell-Weil lattice can realize root systems of types \( E_6 \), \( E_7 \), \( E_8 \) — these have Coxeter numbers 12, 18, 30 respectively, and their fundamental ratios include \( \phi = 1.618 \) (for \( E_8 \), the ratio of largest to smallest root length squared is 2, but the lattice structure yields eigenvalues related to \( 2\cos(\pi/h) \) with \( h=30 \) giving \( 2\cos(\pi/30) \approx 1.989 \), close to 2). The \( E_8 \) root system has 240 roots, and its Weyl group order is 696729600; the lattice is even unimodular, with theta series coefficients involv Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com