2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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
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BibTeX
@misc{caldin20262descent,
title = {{2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23635153},
url = {https://zenodo.org/records/23635153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - 2-Descent via Quartic Models: Torsion Structure Bounds Elliptic Curve Ranks — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23635153
UR - https://zenodo.org/records/23635153
AB - 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
ER -