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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.

##### 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