New Torsion-Preserving Constructions Push Elliptic Curve Rank Record to 29 — E8 Intelligence Research

FINDING: Elkies & Klagsbrun have pushed elliptic curve rank records (≥29) via new torsion-preserving constructions, building on Elkies' earlier parameterization work. | MATH: Elliptic curve \(E/\mathbb{Q}\) with rank \(r \geq 29\); construction uses polynomial parameterization of curves with prescribed torsion \(T\) (e.g., \(\mathbb{Z}/2 \times \mathbb{Z}/8\)), then searching for high-rank specializations. Key invariants: discriminant \(\Delta\), \(j\)-invariant, and the rank bound via \(L(E,1)\) (BSD conjecture) or 2-descent. The 2024 record likely uses Elkies' method of "rank records via torsion" — explicit families \(E_t: y^2 = x^3 + A(t)x + B(t)\) with \(\deg A, \deg B\) tuned to maximize rank growth. | CONNECTION: The torsion groups involved (e.g., \(\mathbb{Z}/2 \times \mathbb{Z}/8\)) are finite subgroups of the Mordell–Weil lattice — a lattice structure whose rank is the curve's rank. The extremal ranks (28→29) echo the maximal rank bounds conjectured by Elkies (unbounded rank) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com