The purpose of this chapter is to identify the GL(2)-Hecke algebras with universal deformation rings with certain additional structures. This fact was first conjectured by B. Mazur and now is a theorem of Wiles in many cases (see Subsection 3.2.7 for a description of the present knowledge to date: October 1999), which is one of the key points of his proof of Fermat's last theorem. In this chapter, we will prove the theorem in a typical case (which covers the case when the weight is bigger than or equal to 2), assuming the knowledge of the modular two-dimensional Galois representations, control theorems of Hecke algebras and the Poitou–Tate duality theorem on Galois cohomology. We will come back later to the duality theorems used here and give a full exposition of them in Chapter 4. As for modular Galois representations and control theorems, we content ourselves only by describing the precise result necessary for the proof and giving some indication of further reading (see Theorem 3.15, Corollary 3.19 and Theorem 3.26). These two results left untouched here will be covered in my forthcoming book [GMF].
Modular Forms on Adele Groups ofGL(2)
We first recall a general theory of elliptic modular forms in the language of adeles.