\(p\)-adic Lie groups.

*(English)*Zbl 1223.22008
Grundlehren der Mathematischen Wissenschaften 344. Berlin: Springer (ISBN 978-3-642-21146-1/hbk; 978-3-642-21147-8/ebook). xi, 254 p. (2011).

In 1965, M. Lazard wrote the seminal paper [“Groupes analytiques \(p\)-adiques”, Publ. Math., Inst. Hautes Étud. Sci. 26, 389–603 (1965; Zbl 0139.02302)] which systematically developed the theory of \(p\)-adic analytic groups. The paper also answered the analogue of Hilbert’s 5th problem for pro-\(p\) groups, viz., “When is it \(p\)-adic analytic?” Apart from answering the basic question above, this paper has such a treasure house of results that until now, perhaps only the results seem to have been well-used and the methods and theory do not seem to have been fully exploited.

The well known notes on “Lie algebras and Lie groups” [Lie algebras and Lie groups. 1964 lectures given at Harvard University. New York-Amsterdam: W. A. Benjamin, Inc. (1965; Zbl 0132.27803)] by J.-P. Serre based on lectures at Harvard University were the main source to learn about \(p\)-adic groups for many years. In 1991, J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal published [Analytic pro-\(p\) groups. London Mathematical Society Lecture Note Series. 157. New York etc.: Cambridge University Press (1991; Zbl 0744.20002)] under the lecture notes series of the London Mathematical Society. This was to understand Lazard’s paper to an extent, viz., to follow the linearity criterion due to A. Lubotzky which was obtained by him using Lazard’s results. This book [DDMS] was more “abstract group-theoretic” in nature as typified by their comment in the introduction: “Starting with powerful pro-\(p\) groups, we could construct most of the group-theoretic consequences of Lazard’s theory without having to introduce any ‘analytic’ machinery. This was a comforting insight for us (as group theorists)”. The approach followed by [DDMS] has paid rich dividends especially in obtaining several results about finite groups apart from abstract profinite groups as well.

In that sense, the book under review complements the earlier book [DDMS] very nicely. In particular, part A of the book thoroughly discusses the analytic aspects of \(p\)-adic manifolds and \(p\)-adic Lie groups. This part, although it may be considered as well known, is put cohesively in textbook style for the first time; this will be very useful for anyone wishing to learn basic \(p\)-adic analysis. This material can possibly be culled out of Bourbaki and Serre’s texts but the advantage here is that, unlike the afore-mentioned pair of texts, the \(p\)-adic case alone is developed instead of having the real case running parallel to it. The author’s text on ‘Nonarchimedean functional analysis’ would also be useful to refer to alongside while reading part A here. One curious point is the use of certain prepositions – the author uses the phrases “the function is continuous in \(x\) or differentiable in \(x\)” while the normal usage is “the function is continuous at \(x\)” etc.!

In part B, the author exposes Lazard’s theory. For those unfamiliar with this subject, I might add that, among other things, Lazard has provided an algebraic approach to studying and characterizing \(p\)-adic analytic groups. For instance, (very) roughly, “a pro-\(p\) group of finite type is \(p\)-adic analytic if the subgroup topologically generated by \(p\)-th powers (\(4\)-th powers if \(p=2\)) of elements contains the commutator subgroup”. Although this assertion is elementary to state, the proof requires a lot of in-depth analysis and new techniques. Lazard had defined a group to be \(p\)-valuable if there is a ‘valuation’ with certain properties and this property proved crucial in obtaining the above-mentioned characterization of \(p\)-adic analyticity (thus answering the analogue of Hilbert’s 5th problem). In [DDMS], the authors did not talk about the notion of \(p\)-valuability but an equivalent notion of “uniformity”. In the present book, the author has followed Lazard’s notion of \(p\)-valuability and described in clear details Lazard’s proof. It is generally opined that Lazard’s paper is not easy to read; thus, this exposition in the book will be extremely useful.

Some of the nice aspects of this (essentially self-contained) book are:

It is not possible in a book of this size to include the important Chapter V of Lazard which discusses cohomology of profinite groups. However, a reference to the book on “Profinite groups” by L. Ribes and P. Zalesskii [Profinite groups. Berlin: Springer (2000; Zbl 0949.20017), 2nd ed. (2010; Zbl 1197.20022)] which discusses those aspects could have been made. Also, the linearity of \(p\)-valuable groups could have been mentioned explicitly.

All in all, this clearly written book by Schneider will be very useful for some years to come, to all those interested in learning the basic theory of \(p\)-adic groups or about the completed group ring of a \(p\)-adic group with number theoretical applications in mind.

The well known notes on “Lie algebras and Lie groups” [Lie algebras and Lie groups. 1964 lectures given at Harvard University. New York-Amsterdam: W. A. Benjamin, Inc. (1965; Zbl 0132.27803)] by J.-P. Serre based on lectures at Harvard University were the main source to learn about \(p\)-adic groups for many years. In 1991, J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal published [Analytic pro-\(p\) groups. London Mathematical Society Lecture Note Series. 157. New York etc.: Cambridge University Press (1991; Zbl 0744.20002)] under the lecture notes series of the London Mathematical Society. This was to understand Lazard’s paper to an extent, viz., to follow the linearity criterion due to A. Lubotzky which was obtained by him using Lazard’s results. This book [DDMS] was more “abstract group-theoretic” in nature as typified by their comment in the introduction: “Starting with powerful pro-\(p\) groups, we could construct most of the group-theoretic consequences of Lazard’s theory without having to introduce any ‘analytic’ machinery. This was a comforting insight for us (as group theorists)”. The approach followed by [DDMS] has paid rich dividends especially in obtaining several results about finite groups apart from abstract profinite groups as well.

In that sense, the book under review complements the earlier book [DDMS] very nicely. In particular, part A of the book thoroughly discusses the analytic aspects of \(p\)-adic manifolds and \(p\)-adic Lie groups. This part, although it may be considered as well known, is put cohesively in textbook style for the first time; this will be very useful for anyone wishing to learn basic \(p\)-adic analysis. This material can possibly be culled out of Bourbaki and Serre’s texts but the advantage here is that, unlike the afore-mentioned pair of texts, the \(p\)-adic case alone is developed instead of having the real case running parallel to it. The author’s text on ‘Nonarchimedean functional analysis’ would also be useful to refer to alongside while reading part A here. One curious point is the use of certain prepositions – the author uses the phrases “the function is continuous in \(x\) or differentiable in \(x\)” while the normal usage is “the function is continuous at \(x\)” etc.!

In part B, the author exposes Lazard’s theory. For those unfamiliar with this subject, I might add that, among other things, Lazard has provided an algebraic approach to studying and characterizing \(p\)-adic analytic groups. For instance, (very) roughly, “a pro-\(p\) group of finite type is \(p\)-adic analytic if the subgroup topologically generated by \(p\)-th powers (\(4\)-th powers if \(p=2\)) of elements contains the commutator subgroup”. Although this assertion is elementary to state, the proof requires a lot of in-depth analysis and new techniques. Lazard had defined a group to be \(p\)-valuable if there is a ‘valuation’ with certain properties and this property proved crucial in obtaining the above-mentioned characterization of \(p\)-adic analyticity (thus answering the analogue of Hilbert’s 5th problem). In [DDMS], the authors did not talk about the notion of \(p\)-valuability but an equivalent notion of “uniformity”. In the present book, the author has followed Lazard’s notion of \(p\)-valuability and described in clear details Lazard’s proof. It is generally opined that Lazard’s paper is not easy to read; thus, this exposition in the book will be extremely useful.

Some of the nice aspects of this (essentially self-contained) book are:

- (i)
- Although \(p\)-adically, a paracompact manifold is topologically a disjoint union of charts, this geometric language is nevertheless persisted with as it is quite intuitive.
- (ii)
- The following refined version proved in the book is pretty useful: any \(p\)-valuable group has a \(p\)-valuation with values in \((1/n) \mathbb{Z}\) for some \(n\).
- (iii)
- Using (ii) above, it is proved that the completed group ring of a pro-\(p\) group of finite rank is Noetherian, regular, of finite global dimension.

It is not possible in a book of this size to include the important Chapter V of Lazard which discusses cohomology of profinite groups. However, a reference to the book on “Profinite groups” by L. Ribes and P. Zalesskii [Profinite groups. Berlin: Springer (2000; Zbl 0949.20017), 2nd ed. (2010; Zbl 1197.20022)] which discusses those aspects could have been made. Also, the linearity of \(p\)-valuable groups could have been mentioned explicitly.

All in all, this clearly written book by Schneider will be very useful for some years to come, to all those interested in learning the basic theory of \(p\)-adic groups or about the completed group ring of a \(p\)-adic group with number theoretical applications in mind.

Reviewer: Balasubramanian Sury (Bangalore)