In addition to providing a self-contained introduction to p-adic lie groups, this volume discusses spaces of locally analytic functions as topological vector spaces, important to applications in representation theory.

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.

First textbook with detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups Treats p-adic analysis in geometric language Includes a chapter on locally convex topology on spaces of locally analytic functions Includes supplementary material: sn.pub/extras

Autorentext
Peter Schneider, geboren 1940 in Lübeck, ist in Süddeutschland aufgewachsen, studierte in Freiburg Germanistik und Geschichte und lebt seit 1961 als freier Schriftsteller. 1972 Staatsexamen, 1973 Berufsverbot als Referendar. Mehrere Förderpreise; 1977/78 Stipendium der Villa Massimo. Schneiders theoretische Schriften dokumentieren den Ablauf der Studentenrevolte der späten 60er Jahre, an der er in Berlin und Italien aktiv teilnahm. Diese Erfahrungen sowie das zeitweilige Berufsverbot bestimmen seine ersten Erzählungen. 2009 erhielt er den Schubart-Literaturpreis der Stadt Aalen.

Inhalt

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 Space C ^an( M,E ), part 1.- II.11 Locally Convex K-V ector Spaces.- II.12 The Topological Vector Space C ^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 of p-A dic Lie Groups.- IV.Preliminaries.- IV.19.Completed Group Rings.- IV.20.The Example of the Group Z ^ 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 Operator P .- V.26.Finite Rank Pro- p -Groups.- V.27.Compact p -Adic Lie Groups.- VI.Completed Group Rings of p -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.Rational p -Valuations and Applications.- VII.34.Coordinates of the First and of the Second Kind.- References.- Index.