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p-Adic Lie Groups

:	Peter Schneider
:	p-Adic Lie Groups
:	Springer-Verlag
:	9783642211478
:	1
:	CHF 104.40
:

:	Arithmetik, Algebra
:	English

:	259
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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.

	Introduction	6
	Contents	9
	Part A p-Adic Analysis and Lie Groups	12
	I Foundations	13
	1 Ultrametric Spaces	13
	2 Nonarchimedean Fields	18
	3 Convergent Series	24
	4 Differentiability	27
	5 Power Series	35
	6 Locally Analytic Functions	48
	II Manifolds	54
	7 Charts and Atlases	54
	8 Manifolds	56
	9 The Tangent Space	65
	10 The Topological Vector Space Can (M, E), Part 1	83
	11 Locally Convex K-Vector Spaces	88
	12 The Topological Vector Space Can (M, E), Part 2	93
	III Lie Groups	98
	13 Definitions and Foundations	98
	14 The Universal Enveloping Algebra	110
	15 The Concept of Free Algebras	115
	16 The Campbell-Hausdorff Formula	120
	17 The Convergence of the Hausdorff Series	133
	18 Formal Group Laws	141
	Part B The Algebraic Theory of p-Adic Lie Groups	163
	IV Preliminaries	164
	19 Completed Group Rings	164
	20 The Example of the Group Zpd	170
	21 Continuous Distributions	171
	22 Appendix: Pseudocompact Rings	172
	V p-Valued Pro-p-Groups	175
	23 p-Valuations	175
	24 The Free Group on Two Generators	181
	25 The Operator P	184
	26 Finite Rank Pro-p-Groups	187
	27 Compact p-Adic Lie Groups	198
	VI Completed Group Rings of p-Valued Groups	201
	28 The Ring Filtration	201
	29 Analyticity	207
	30 Saturation	214
	VII The Lie Algebra	224
	31 A Normed Lie Algebra	224
	32 The Hausdorff Series	237
	33 Rational p-Valuations and Applications	248
	34 Coordinates of the First and of the Second Kind	252
	References	256
	Index	258