2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research

FINDING: Complete 2-descent on elliptic curves over ℚ reduces the Mordell-Weil rank problem to finite checks on quartic (degree-4) models, linking torsion structure to rational point lattices. | MATH: For E: y² = x³ + ax + b, 2-descent maps E(ℚ)/2E(ℚ) ↪ Sel⁽²⁾(E/ℚ), with quartic models C: y² = f(x) where deg(f)=4 (or genus-1 binary quartics). The 2-torsion E[2] ≅ (ℤ/2ℤ)² over ℚ determines the factorization of f into quadratics; the Selmer group bounds rank r ≤ dim₂ Sel⁽²⁾ − dim₂ E[2](ℚ). Yesin's work: divisibility by 2 on quartic models connects to rational Diophantine D(q)-quintuples — sets {a₁,…,a₅} where aᵢaⱼ + q is a square for all i≠j, with q ∈ ℚ*. | CONNECTION: The quartic model's binary form has discriminant Δ = 16·disc(f). The torsion lattice E[2] corresponds to the root system A₁³ (three orthogonal ℤ/2ℤ axes) — a crystallographic symmetry. The D(q)-quintuple condition aᵢaⱼ + q = square is structurally identical to the condition for rational points on a genus-1 curve, whose j-i

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com