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p-Adic Lie Groups

Peter Schneider
Publication Date: 
Number of Pages: 
Grundlehren der mathematischen Wissenschaften 344
[Reviewed by
Fernando Q. Gouvêa
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The notion of a p-adic Lie group has been around for a while, but they have recently become more prominent in number theory and representation theory. Unfortunately, up to now the only references on the subject were fairly hard going: Serre's notes and Bourbaki's encyclopedic treatment of Lie theory both included the p-adic case, but neither was particularly accessible; Lazard's famous paper was notoriously hard to read. Schneider's Grundlehren volume is an attempt to fill that gap by giving a systematic treatment of the subject.

The book has two very different parts. The first, based on a graduate course Schneider teaches at Münster, gives a p-adic-analytic account of the theory which is much like the classical theory of Lie groups: a p-adic Lie group is a (locally analytic) manifold over a non-archimedean field that has a locally analytic group structure. The treatment is in the classic "knapp aber lückenloss " style: no words are wasted, but proofs are complete. The reader has to pay attention. For example, the word "manifold" is quickly restricted to "locally analytic manifold over a non-archimedean field", so one finds theorems that say things like "any paracompact manifold is the disjoint union of open charts." This is, of course, completely false in the archimedean case, and it shows that (as Schneider puts it in the introduction) the notion of a p-adic Lie group "has no geometric content."

This part of the book should be accessible (given a willingness to read carefully with pencil in hand and a good library nearby) to anyone who knows classical Lie theory and basic p-adic analysis. In particular, it does not require rigid analytic geometry, which left me wondering what a theory of rigid-analytic Lie groups would be like.

The second part of the book, based on a course given at the Newton Institute, develops further the theme of "no geometric content" by giving a detailed account of Lazard's purely algebraic theory of p-adic Lie groups. Lazard's paper is famously hard to read, so Schneider's exposition is to be welcomed. This remains advanced material, however, and "easier" does not mean "easy".

Overall, this is a book to be welcomed and studied carefully by anyone who wants to learn about p-adic Lie theory.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.