Modular forms and Galois cohomology.

*(English)*Zbl 0952.11014
Cambridge Studies in Advanced Mathematics. 69. Cambridge: Cambridge University Press. x, 343 p. (2000).

In the late 80’s, Mazur conjectured that \(p\)-adic Hecke algebras are, under some precise conditions, universal deformation rings of two-dimensional Galois representations. Many cases of Mazur’s conjecture have been proved by Wiles. He then went on to prove the Shimura-Taniyama-Weil conjecture for many elliptic curves over \(\mathbb{Q}\), including semistable ones, and pulled Fermat’s Last Theorem out of the conjecture [A. Wiles, Ann. Math. (2) 141, 443-551 (1995; Zbl 0823.11029), R. Taylor and A. Wiles [Ann. Math. (2) 141, 553-572 (1995; Zbl 0823.11030)].

The present monograph is based on several advanced courses given by the author on the main parts of the proof by Wiles (and Taylor) of Fermat’s Last Theorem. Special attention is given to provide an identification of certain Hecke algebras with universal deformation rings of (2-dimensional) Galois representations (Chapter 3). The exposition includes several improvements and simplifications in the original proof, due to F. Diamond and K. Fujiwara. The material covered by the first four chapters may certainly serve as a systematic introduction to the original Wiles’s article (and covers it to some extent). The final chapter 5 contains new results (due to the author) concerning the base-change of deformation rings, \(p\)-adic Hecke algebras and computing the Selmer groups attached to adjoint modular Galois representations.

Here is a summary of this monograph.

Chapter 1 contains a brief overview of the theory of modular forms. A general theory of group representations, pseudorepresentations, and their deformations is the theme of Chapter 2. General notions (profinite group, induced representation, …) as well as basis results (density theorem, Brauer-Nesbitt theorem) are summarized in Sections 2.1.1-2.1.6. Section 2.1.7 contains basis results (due to Mazur and Carayol-Serre) concerning representations with coefficients in Artinian rings. The trace of a representations is characterized in Section 2.2. The last section of Chapter 2 is a brief introduction to deformation theory of group representations. The purpose of Chapter 3 is to identify the \(GL(2)\)-Hecke algebras with universal deformation rings with certain additional structures (Sections 3.2.6, 3.2.7, 3.2.8). Full proofs are given, assuming the knowledge of the modular 2-dimensional Galois representations, control theorems of Hecke algebras and the Poitou-Tate duality results on Galois cohomology. The first two mentioned results are only formulated (Theorems 3.15 and 3.26). Full exposition of the Poitou-Tate duality theorem and the Euler characteristic formulas is postponed to Chapter 4. In the last Chapter 5, the author studies adjoint Galois representations of a given Galois representation. The main results of this chapter include: Nonabelian adjoint version of the analytic class number formula (Theorem 5.20), control theorem of the adjoint Selber groups (Theorem 5.42), and the torsionness of the adjoint Selmer group over the infinite cyclotomic extension (Theorem 5.44).

The present monograph is based on several advanced courses given by the author on the main parts of the proof by Wiles (and Taylor) of Fermat’s Last Theorem. Special attention is given to provide an identification of certain Hecke algebras with universal deformation rings of (2-dimensional) Galois representations (Chapter 3). The exposition includes several improvements and simplifications in the original proof, due to F. Diamond and K. Fujiwara. The material covered by the first four chapters may certainly serve as a systematic introduction to the original Wiles’s article (and covers it to some extent). The final chapter 5 contains new results (due to the author) concerning the base-change of deformation rings, \(p\)-adic Hecke algebras and computing the Selmer groups attached to adjoint modular Galois representations.

Here is a summary of this monograph.

Chapter 1 contains a brief overview of the theory of modular forms. A general theory of group representations, pseudorepresentations, and their deformations is the theme of Chapter 2. General notions (profinite group, induced representation, …) as well as basis results (density theorem, Brauer-Nesbitt theorem) are summarized in Sections 2.1.1-2.1.6. Section 2.1.7 contains basis results (due to Mazur and Carayol-Serre) concerning representations with coefficients in Artinian rings. The trace of a representations is characterized in Section 2.2. The last section of Chapter 2 is a brief introduction to deformation theory of group representations. The purpose of Chapter 3 is to identify the \(GL(2)\)-Hecke algebras with universal deformation rings with certain additional structures (Sections 3.2.6, 3.2.7, 3.2.8). Full proofs are given, assuming the knowledge of the modular 2-dimensional Galois representations, control theorems of Hecke algebras and the Poitou-Tate duality results on Galois cohomology. The first two mentioned results are only formulated (Theorems 3.15 and 3.26). Full exposition of the Poitou-Tate duality theorem and the Euler characteristic formulas is postponed to Chapter 4. In the last Chapter 5, the author studies adjoint Galois representations of a given Galois representation. The main results of this chapter include: Nonabelian adjoint version of the analytic class number formula (Theorem 5.20), control theorem of the adjoint Selber groups (Theorem 5.42), and the torsionness of the adjoint Selmer group over the infinite cyclotomic extension (Theorem 5.44).

Reviewer: Andrzej Dąbrowski (Szczecin)

##### MSC:

11F75 | Cohomology of arithmetic groups |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11F11 | Holomorphic modular forms of integral weight |

11F80 | Galois representations |