Tilouine, Jacques 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. Reviewer: Ian Kiming (MR 99i:11038) Cited in 2 ReviewsCited in 18 Documents 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 Keywords:universal nearly ordinary Hecke algebra; nearly ordinary representations; Krull dimension; Galois cohomology; Hida-Iwasawa algebra PDFBibTeX XMLCite \textit{J. Tilouine}, Deformations of Galois representations and Hecke algebras. New Delhi: Narosa Publishing House; Publ. for the Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad (1996; Zbl 1009.11033)