Deformations of Galois representations and Hecke algebras.

*(English)*Zbl 1009.11033
New Delhi: Narosa Publishing House; Publ. for the Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad. xii, 108 p. (1996).

This book is a very nice and compact introduction to the topics of the title. In the first part—Chapters 1 to 5—the author develops a general basic theory of deformations of Galois representations in the following setting: \(p\) is a prime number, \({\mathfrak O}\) is the ring of integers in a finite extension of \(\mathbb{Q}_p\) with residue field \(k\), \(G\) is a smooth linear group\(/{\mathfrak O}\), \(F\) is a number field, \(S\) is a finite set of primes of \(F\) including the Archimedean ones and those lying over \(p\), \(F_S\) is the maximal Galois extension of \(F\) unramified outside \(S\) inside \(\overline{F} \subseteq \mathbb{C}\), and we are given a continuous representation \(\overline{\rho} \colon\text{Gal}(F_S/F) \rightarrow G(k)\). A sufficient condition for the existence of a universal deformation is found in Chapter 3.

In the following Chapters 6-9, the theory is specialized to representations \(\overline{\rho}\) which are nearly ordinary; here, “nearly ordinary” is defined as the property that for each prime \(v\) over \(p\), \(\overline{\rho}\) maps a fixed decomposition group at \(v\) into \(P_v(k)\), where \(P_v\) is a fixed parabolic subgroup of \(G\). One has a similar notion of nearly ordinary deformation of \(\overline{\rho}\). Sufficient conditions for the existence of a universal nearly ordinary deformation are given. The subsequent discussions then center around the question of determining the Krull dimension of the universal nearly ordinary deformation ring \(R^{no}\) when it exists. Under some additional conditions, a precise conjecture expressing the dimension of \(\mathbb{Q}_p \otimes R^{no}\) by means of Galois cohomology is formulated on page 72. Chapter 8 is devoted to the introduction of a Hida-Iwasawa algebra \(\Lambda\) attached to \(G\) and the given collection \(P_v\) of parabolic subgroups, and to the construction of a structure on \(R^{no}\) as a \(\Lambda\)-algebra. Also, the dimension of \(\Lambda\) over \({\mathfrak O}\) is studied. The book culminates in Chapter 10 with the construction of the universal nearly ordinary Hecke algebra and the formulation of conjectures which—under certain conditions—would have \(R^{no}\) for a \(\overline{\rho}\) “coming from” a cuspidal algebraic automorphic form isomorphic as a \(\Lambda\)-algebra to a local component of the universal nearly ordinary Hecke algebra.

A very attractive feature of the book is the inclusion of numerous and detailed discussions of particular examples illustrating the general theory.

In the following Chapters 6-9, the theory is specialized to representations \(\overline{\rho}\) which are nearly ordinary; here, “nearly ordinary” is defined as the property that for each prime \(v\) over \(p\), \(\overline{\rho}\) maps a fixed decomposition group at \(v\) into \(P_v(k)\), where \(P_v\) is a fixed parabolic subgroup of \(G\). One has a similar notion of nearly ordinary deformation of \(\overline{\rho}\). Sufficient conditions for the existence of a universal nearly ordinary deformation are given. The subsequent discussions then center around the question of determining the Krull dimension of the universal nearly ordinary deformation ring \(R^{no}\) when it exists. Under some additional conditions, a precise conjecture expressing the dimension of \(\mathbb{Q}_p \otimes R^{no}\) by means of Galois cohomology is formulated on page 72. Chapter 8 is devoted to the introduction of a Hida-Iwasawa algebra \(\Lambda\) attached to \(G\) and the given collection \(P_v\) of parabolic subgroups, and to the construction of a structure on \(R^{no}\) as a \(\Lambda\)-algebra. Also, the dimension of \(\Lambda\) over \({\mathfrak O}\) is studied. The book culminates in Chapter 10 with the construction of the universal nearly ordinary Hecke algebra and the formulation of conjectures which—under certain conditions—would have \(R^{no}\) for a \(\overline{\rho}\) “coming from” a cuspidal algebraic automorphic form isomorphic as a \(\Lambda\)-algebra to a local component of the universal nearly ordinary Hecke algebra.

A very attractive feature of the book is the inclusion of numerous and detailed discussions of particular examples illustrating the general theory.

Reviewer: Ian Kiming (MR 99i:11038)

##### MSC:

11F80 | Galois representations |

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

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

11R34 | Galois cohomology |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |