p-Adic Lie Groups by Peter Schneiderp-Adic Lie Groups by Peter Schneider

p-Adic Lie Groups

byPeter Schneider

Paperback | August 3, 2013

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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.
Title:p-Adic Lie GroupsFormat:PaperbackDimensions:256 pages, 23.5 × 15.5 × 0.02 inPublished:August 3, 2013Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3642268668

ISBN - 13:9783642268663


Table of Contents

Introduction.-Part A :p-Adic Analysis and Lie Groups.- I.Foundations.- I.1.Ultrametric Spaces.- I.2.Nonarchimedean Fields.- I.3.Convergent Series.- I.4.Differentiability.- I.5.Power Series.- I.6.Locally Analytic Functions.- II.Manifolds.- II.7.Charts and Atlases.- II.8.Manifolds.- II.9.The Tangent Space.- II.10.The Topological Vector SpaceC^an(M,E), part 1.- II.11 Locally ConvexK-Vector Spaces.- II.12 The Topological Vector SpaceC^an(M,E), part 2.- III.Lie Groups.- III.13.Definitions and Foundations.- III.14.The Universal Enveloping Algebra.- III.15.The Concept of Free Algebras.- III.16.The Campbell-Hausdorff Formula.- III.17.The Convergence of the Hausdorff Series.- III.18.Formal Group Laws.-Part B: The Algebraic Theory ofp-Adic Lie Groups.- IV.Preliminaries.- IV.19.Completed Group Rings.- IV.20.The Example of the GroupZ ^d_p.- IV.21.Continuous Distributions.- IV.22.Appendix: Pseudocompact Rings.- V.p-Valued Pro-p-Groups.- V.23.p-Valuations.- V.24.The free Group on two Generators.- V.25.The OperatorP.- V.26.Finite Rank Pro-p-Groups.- V.27.Compactp-Adic Lie Groups.- VI.Completed Group Rings ofp-Valued Groups.- VI.28.The Ring Filtration.- VI.29.Analyticity.- VI.30.Saturation.- VII.The Lie Algebra.- VII.31.A Normed Lie Algebra.- VII.32.The Hausdorff Series.- VII.33.Rationalp-Valuations and Applications.- VII.34.Coordinates of the First and of the Second Kind.- References.- Index.

Editorial Reviews

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)