The Frey Curve: The Bridge Between Fermat's Last Theorem and Modularity — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The Frey curve construction — a semistable elliptic curve with discriminant linked to a hypothetical Fermat counterexample — is the pivotal bridge forcing the contradiction between modularity and level lowering, completing the FLT proof. | MATH: Frey curve: Ea,b,c: y2 = x(x−ap)(x+bp) for ap + bp = cp, p ≥ 5 prime. Discriminant Δ = (abc)2p · 16 (up to sign), conductor N = rad(abc) (squarefree — semistable). Ribet's level lowering: if E is modular of level N, then a newform of level 2 exists (since p | Δ with p ≥ 5 and p nmid N), but no elliptic curves over ℚ have conductor 2 (or any level < 11). Modularity (Taniyama–Shimura–Weil, proved by Wiles–Taylor–Breuil–Conrad–Diamond–Taylor): every semistable elliptic curve over ℚ is modular. Contradiction: E cannot exist → FLT true. | CONNECTION: The Frey curve's j-invariant: \(
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026frey,
title = {{The Frey Curve: The Bridge Between Fermat's Last Theorem and Modularity — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23447118},
url = {https://zenodo.org/records/23447118},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Frey Curve: The Bridge Between Fermat's Last Theorem and Modularity — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23447118
UR - https://zenodo.org/records/23447118
AB - FINDING: The Frey curve construction — a semistable elliptic curve with discriminant linked to a hypothetical Fermat counterexample — is the pivotal bridge forcing the contradiction between modularity and level lowering, completing the FLT proof. | MATH: Frey curve: E_a,b,c: y^2 = x(x−a^p)(x+b^p) for a^p + b^p = c^p, p ≥ 5 prime. Discriminant Δ = (abc)^2p · 16 (up to sign), conductor N = rad(abc) (squarefree — semistable). Ribet's level lowering: if E is modular of level N, then a newform of level 2 exists (since p | Δ with p ≥ 5 and p nmid N), but no elliptic curves over ℚ have conductor 2 (or any level < 11). Modularity (Taniyama–Shimura–Weil, proved by Wiles–Taylor–Breuil–Conrad–Diamond–Taylor): every semistable elliptic curve over ℚ is modular. Contradiction: E cannot exist → FLT true. | CONNECTION: The Frey curve's j-invariant: \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -