The Frey Curve Bridge: Linking Fermat's Last Theorem to Modularity — E8 Intelligence Research

Abstract

FINDING: The Frey curve construction links Fermat's Last Theorem to modularity via Galois representations, with the discriminant Δ = (abc)^{2p} forcing a semistable elliptic curve whose 3-adic and p-adic Galois images are constrained by Serre's conjecture. | MATH: Frey curve: y² = x(x − aᵖ)(x + bᵖ); discriminant Δ = (abc)^{2p} / 256; conductor N = rad(abc) (squarefree, semistable); Galois representation ρ_{E,p}: Gal(ℚ̄/ℚ) → GL₂(𝔽ₚ); modular form weight 2, level N; Ribet's level-lowering: ρ_{E,p} ≅ ρ_{f,𝔭} for f of weight 2, level 2 (impossible — no such cusp form exists). | CONNECTION: The Frey curve's discriminant is a perfect 2p-th power — the exponent 2p creates a ratio structure where the j-invariant j(E) = 256(a^{2p} + b^{2p})³/(abc)^{2p} scales with (abc)^{-2p}, echoing the 0.618/1.618 golden ratio inversions in the sense that the curve's arithmetic complexity (conductor) is minimized (squarefree) while its discriminant is maximally powered — a dual extremality. The semistable re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com