Kisin's deformation-ring framework extends modularity lifting beyond Taylor–Wiles — E8 Intelligence Research

FINDING: Modularity lifting has been extended beyond the Taylor–Wiles hypothesis via Kisin's deformation-ring framework, with new results for non-regular symplectic representations and defect-zero deformation rings. | MATH: Core object is the Galois deformation ring R and its map R → T (Hecke algebra); Taylor–Wiles patching produces a finite free module over a power series ring S_∞ = W[[y₁,…,y_q]] with q = dim of a Selmer/cohomology group; Kisin's method replaces patching with the study of the *framed* deformation ring and its relation to the unrestricted one via a "Kisin resolution" of length ≤ 1. Key numerical invariant: the *defect* = dim H¹ − dim H² (or the difference between expected and actual dimension of the deformation ring). Taylor–Wiles hypothesis requires ℓ ≡ 1 mod p and the local representation at ℓ to be a sum of distinct characters; Kisin's method removes this by allowing ℓ = p and non-regular weights. | CONNECTION: The patching construction is a *lattice-theoretic* oper Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com