Frey's Curve: Linking FLT Counterexamples to Modularity Violations — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Frey's curve construction shows a FLT counterexample (a^p + b^p = c^p) would yield a semistable elliptic curve with discriminant a perfect p-th power, violating the Taniyama-Shimura (modularity) theorem — the core of Wiles' proof. | MATH: Frey curve: y² = x(x - a^p)(x + b^p); discriminant Δ = (abc)^(2p) / 2^8 (up to sign and 2-adic factors); conductor N is squarefree (semistable) with specific primes; Ribet's level-lowering: if modular of level N, then modular of level 2 — impossible for p ≥ 3 due to weight-2 cusp form dimension zero at level 2. Key constants: p ≥ 3 prime; the curve has multiplicative reduction at primes dividing abc; j-invariant = 2^8 (a^(2p) + b^(2p) + ... )^3 / (abc)^(2p) — denominator is a perfect p-th power. | CONNECTION: The discriminant's exponent 2p and the conductor's squarefree nature force a rigid arithmetic structure; the modular form's weight 2 and level 2 relate to the elliptic point of order 2 in the modular curve X₀(2) — a lattice symmetry (the
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026freys,
title = {{Frey's Curve: Linking FLT Counterexamples to Modularity Violations — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23464938},
url = {https://zenodo.org/records/23464938},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Frey's Curve: Linking FLT Counterexamples to Modularity Violations — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23464938
UR - https://zenodo.org/records/23464938
AB - FINDING: Frey's curve construction shows a FLT counterexample (a^p + b^p = c^p) would yield a semistable elliptic curve with discriminant a perfect p-th power, violating the Taniyama-Shimura (modularity) theorem — the core of Wiles' proof. | MATH: Frey curve: y² = x(x - a^p)(x + b^p); discriminant Δ = (abc)^(2p) / 2^8 (up to sign and 2-adic factors); conductor N is squarefree (semistable) with specific primes; Ribet's level-lowering: if modular of level N, then modular of level 2 — impossible for p ≥ 3 due to weight-2 cusp form dimension zero at level 2. Key constants: p ≥ 3 prime; the curve has multiplicative reduction at primes dividing abc; j-invariant = 2^8 (a^(2p) + b^(2p) + ... )^3 / (abc)^(2p) — denominator is a perfect p-th power. | CONNECTION: The discriminant's exponent 2p and the conductor's squarefree nature force a rigid arithmetic structure; the modular form's weight 2 and level 2 relate to the elliptic point of order 2 in the modular curve X₀(2) — a lattice symmetry (the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -