The deformation theoretic techniques of Wiles-Taylor were introduced for elliptic modular forms in the introductory Chapter 1, and are generalized to Hilbert modular forms (following Fujiwara's treatment) in this chapter. In particular, Fujiwara's ‘R=T’ theorem (the identification of the Galois deformation ring and the corresponding Hecke algebra) is proven in the minimal case. In addition to the Taylor-Wiles methods, an explicit formula of the L-invariant (of the adjoint L-functions) as well as an integral solution to Eichler's basis problem are presented for Hilbert modular forms.
Keywords: R=T theorem, Taylor-Wiles system, Adjoint Selmer group, near ordinarity