Deformation Theory of Galois Representations and the Taylor--Wiles Method
Ehsan Shahoseini
TL;DR
This chapter gives a concise roadmap of the Taylor--Wiles patching method by grounding it in Mazur's deformation theory of Galois representations, detailing both local and global deformation problems and the construction of universal deformation rings. It develops the notion of framed and unframed deformations, ties tangent spaces to group cohomology, and explains the role of $p$-finiteness and representability criteria, setting the stage for introducing Taylor--Wiles primes and the patching mechanism. The core development builds augmented global deformation problems with Taylor--Wiles data, constructs the patched objects $R_\infty$ and $M_\infty$, and proves $R=\mathbb{T}$ in minimal and non-minimal cases via Diamond's patching framework. Finally, it connects the Galois side to automorphic data through Hecke algebras and Eichler--Shimura theory, outlining modularity lifting theorems under suitable hypotheses and the analytic payoff for Shimura--Taniyama--Weil type conjectures.
Abstract
In this chapter, we want to have an overview of the Taylor--Wiles patching method. For this purpose, at the first, we recall Mazur's theory of deforming Galois representations and study both local and global deformation problems. Then, we go through the subject of Taylor-Wiles primes and examine the role that they play on the Galois side and the modular (automorphic) side. At the end, we arrive at the Taylor-Wiles patching method and use it to prove $R=\mathbb{T}$ in both minimal and non-minimal cases. Note that, in the Galois side, we will work with totally real number fields, but for the modular side, we will concentrate on $\mathbb{Q}$ to avoid difficulties of working with Hilbert modular forms.