p-Adic Lie Groups
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 introduc
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Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin S. Mori B.C. Ngô M. Ratner D. Serre N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan
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Peter Schneider
p-Adic Lie Groups
Peter Schneider Institute of Mathematics University of Münster Einsteinstrasse 62 Münster 48149 Germany [email protected]
ISSN 0072-7830 ISBN 978-3-642-21146-1 e-ISBN 978-3-642-21147-8 DOI 10.1007/978-3-642-21147-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011930424 Mathematics Subject Classification: 22E20, 16S34 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Introduction This book presents a complete account of the foundations of the theory of p-adic Lie groups. It moves on to some of the important more advanced aspects. Although most of the material is not new, it is only in recent years that p-adic Lie groups have found important applications in number theory and representation theory. These applications constitute, in fact, an increasingly active area of research. The book is designed to give to the advanced, but not necessarily graduate, student a streamlined access to the basics of the theory. It is almost self contained. Only a few technical computations which are well covered in the literature will not be repeated. My hope is that researchers who see the need to take up p-adic methods also will find this book helpful for quickly mastering the necessary notions and techniques. The book comes in two parts. Part A on the analytic side grew out of a course which I gave at M¨ unster for the first time during the summer term 2001, whereas part B on the algebraic side is the content of a course given at the Newton Institute during September 2009. The original and proper context of p-adic Lie groups is p-adic analysis. This is the point of view in Part A. Of course, in a formal sense the notion of a p-adic Lie group is completely parallel to the classical notion of a real or com
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