Geometric Methods in the Algebraic Theory of Quadratic Forms Summer
The geometric approach to the algebraic theory of quadratic forms is the study of projective quadrics over arbitrary fields. Function fields of quadrics have been central to the proofs of fundamental results since the renewal of the theory by Pf
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Oleg T. Izhboldin Bruno Kahn Nikita A. Karpenko Alexander Vishik
Geometric Methods in the Algebraic Theory of Quadratic Forms Summer School, Lens, 2000 Editor: Jean-Pierre Tignol
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Authors Oleg T. Izhboldin (Deceased April 17, 2000) Bruno Kahn Institut de Math´ematiques de Jussieu 175-179 rue du Chevaleret 75013 Paris, France
Alexander Vishik Institute for Information Transmission Problems Russian Academy of Sciences Bolshoj Karetnyj Pereulok, Dom 19 101447 Moscow, Russia e-mail: [email protected]
e-mail: [email protected]
Nikita A. Karpenko Universit´e d’Artois Rue Jean Souvraz SP 18 62307 Lens, France e-mail: [email protected]
Editor Jean-Pierre Tignol Institut de Math´ematique Pure et Appliqu´ee Universit´e catholique de Louvain Chemin du Cyclotron 2 1348 Louvain-la-Neuve, Belgium e-mail: [email protected]
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 11E81, 14C15, 14F42 ISSN 0075-8434 ISBN 3-540-20728-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10976173
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In memory of Oleg Tomovich Izhboldin (1963–2000)
Preface
The geometric approach to the algebraic theory of quadratic forms is the study of projective quadrics over arbitrary fields. Function fields of quadrics were a basic ingredient already in the proof of the Arason–Pfister Hauptsatz of 1971 (or even in Pfister’s 1965 construction of fields with prescribed level); they are central in the investigation of deep properties of quadratic forms, such as their splitting pattern, but also in the construction of fields which exhibit particular properties, such as a given u-invariant. Recently, finer geometric tools have
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