Worked solution: Wiles's modularity proof via the Taylor–Wiles method (1994)

Wiles announced a complete proof in three lectures at the Isaac Newton Institute in Cambridge on June 21–23, 1993, based on an extension of a construction of Matthias Flach's to bound the Selmer group controlling $R_{\Sigma}$ via an Euler system. As Wiles recounts, 'it became clear to me in the fall of 1993 that the construction of the Euler system used to extend Flach's method was incomplete and possibly flawed' (Wiles 1995, p. 453). Nick Katz, who had carefully read the draft, discovered the specific gap, which lay entirely inside the bound on the Selmer group needed to show $R_{\Sigma}$ and $\mathbb{T}_{\Sigma}$ have the same size, not in Frey's construction (Step 2) or Ribet's theorem (Step 6), both of which were correct from the start.

Richard Taylor joined Wiles in January 1994 to try to repair the Euler system argument; after this failed, they attempted a new approach using $p = 2$ in spring 1994, which reached an impasse by the end of August 1994 (Wiles 1995, p. 453). In September 1994, Wiles returned to an idea from the summer of 1991 involving auxiliary primes $q_i \equiv 1 \bmod p^{n_i}$ with $n_i \to \infty$, which he had abandoned when he thought the Euler system approach was correct. On September 19, 1994, drawing on a construction of Ehud de Shalit for primes $q \equiv 1 \bmod p$, Wiles 'saw in a flash' that Hecke rings at these auxiliary levels could be glued together via Galois cohomology duality into a single power series ring, giving precisely the required bound on the Selmer group without any Euler system.

This patching technique, the Taylor–Wiles method, was worked out jointly and published as Taylor and Wiles, 'Ring-theoretic properties of certain Hecke algebras,' Annals of Mathematics 141 (1995), 553–572, in the same volume as Wiles's paper. Together, they establish $R_{\Sigma} \cong \mathbb{T}_{\Sigma}$ for the minimal case with $p = 3$ (or $p = 5$ via Step 8), completing the modularity lifting argument and proving Wiles's Theorem 0.4: every semistable elliptic curve over $\mathbb{Q}$ is modular.