p-Adic Lie Groups

1. Front Matter

   Pages I-XI

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2. p-Adic Analysis and Lie Groups

   1. Front Matter

      Pages 1-1

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3. $p$-adic analysis and Lie groups

   1. Foundations

      • Peter Schneider

      Pages 3-43

   2. Manifolds

      • Peter Schneider

      Pages 45-88

   3. Lie Groups

      • Peter Schneider

      Pages 89-153

4. The Algebraic Theory of p-Adic Lie Groups

   1. Front Matter

      Pages 155-155

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5. The algebraic theory of $p$-adic Lie groups

   1. Preliminaries

      • Peter Schneider

      Pages 157-167

   2. p-Valued Pro-p-Groups

      • Peter Schneider

      Pages 169-194

   3. Completed Group Rings of p-Valued Groups

      • Peter Schneider

      Pages 195-217

   4. The Lie Algebra

      • Peter Schneider

      Pages 219-250

6. Back Matter

   Pages 251-254

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Manifolds over complete nonarchimedean fields together with notions like tangent spaces and vector fields form a convenient geometric language to express the basic formalism of p-adic analysis. The volume starts with a self-contained and detailed introduction to this language. This includes the discussion of spaces of locally analytic functions as topological vector spaces, important for applications in representation theory. The author then sets up the analytic foundations of the theory of p-adic Lie groups and develops the relation between p-adic Lie groups and their Lie algebras. The second part of the book contains, for the first time in a textbook, a detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups, via his notion of a p-valuation, together with its application to the structure of completed group rings.

• 22E20, 16S34
• Lie group
• completed group ring
• locally analytic manifold
• p-adic
• p-valuation

From the reviews:

“The book is divided into two parts … . The author’s style of writing is elegant … . this is a demanding book, but a rewarding one … . any person who intends to work in this area will want to have it close at hand.” (Mark Hunacek, The Mathematical Gazette, Vol. 98 (541), March, 2014)

“This book presents the foundations of the theory of p-adic Lie groups in a systematic and self-contained way. … Schneider’s book on p-adic Lie groups systematically develops the analytic theory of p-adic Lie groups and also Lazard’s algebraic approach to p-adic Lie groups. It is highly recommended.” (Dubravka Ban, Mathematical Reviews, Issue 2012 h)

“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. … Schneider’s Grundlehren volume is an attempt to fill that gap by giving a systematic treatment of the subject. … this is a book to be welcomed and studied carefully by anyone who wants to learn about p-adic Lie theory.” (Fernando Q. Gouvêa, The Mathematical Association of America, August, 2011)

“The book thoroughly discusses the analytic aspects of p-adic manifolds and p-adic lie groups. … this clearly written book by Schneider will be very useful … to all those interested 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.” (Bala Sury, Zentralblatt MATH, Vol. 1223, 2011)

• , Institute of Mathematics, University of Münster, Münster, Germany

  Peter Schneider

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