Fractal-Torsional-Spectral Extensions to Wiles' Modularity Theorem

Wiles’ modularity theorem establishes that every semistable elliptic curve E/Q is modular, meaning its L-function coincides with that of a modular form f: L(s, E) = L(s, f). This result is achieved via a deformation-theoretic framework that links the Galois representation ρE, : Gal(Q/Q) → GL2(Z) to Hecke algebras through the R = T isomorphism. The proof employs techniques from commutative algebra, deformation theory, and the introduction of Taylor-Wiles primes to control Selmer groups. A fundamental consequence is the resolution of Fermat’s Last Theorem, as the modularity of the Frey elliptic curve contradicts prior results, eliminating any possible counterexamples.