Complete 2-Descent via Quartic Models for Mordell-Weil and Selmer Groups — E8 Intelligence Research

FINDING: Complete 2-descent computes the Mordell-Weil group E(Q) via the exact sequence 0 → E(Q)/2E(Q) → Sel²(E/Q) → Sha(E/Q)[2] → 0, with quartic models providing the explicit computational realization. | MATH: For E: y² = x³ + Ax + B, a 2-cover is a quartic Y² = aX⁴ + bX² + c; the 2-Selmer group is cut out by local solubility conditions at all places; E(Q)/2E(Q) ≅ image of Sel² under the descent map; rank r = dim_{F₂}(Sel²) − dim_{F₂}(Sha[2]) − 2 (for full 2-torsion). Torsion subgroup E(Q)_tors ∈ {Z/nZ : n ≤ 10, n=12} ∪ {Z/2Z × Z/2nZ : n ≤ 4} (Mazur). | CONNECTION: The quartic model Y² = quartic(X) is a genus-1 curve; its Jacobian is E. The 2-torsion lattice E[2] ≅ (Z/2)² sits inside the Mordell-Weil lattice E(Q) with the Néron-Tate height pairing — a positive-definite lattice structure. Rank-29 curve (Elkies–Klagsbrun 2024) pushes the lattice rank record; no geometric-harmony constants (0.618, 1.618, base-60) appear intrinsically in 2-descent — the relevant arithmetic is F₂-linear a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com