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CONTEMPORARY MATHEMATICS
272
Quadratic Forms and Their Applications
Proceedings of the Conference on Quadratic Forms and Their Applications
July 5-9, 1999 University College Dublin
Eva Bayer-Fiuckiger, David Lewis, Andrew Ranicki, Editors
Selected Titles in This Series
272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic forms and their applications, 2000
271 J.P.C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy methods in algebraic topology, 2000
270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000
269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in geometry, topology and dynamics, 2001
268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000
267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000
266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000
265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000
264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000
263 Yan Guo, Editor, Nonlinear wave equations, 2000 262 Paul Igodt, Herbert Abels, Yves Felix, and Fritz Grunewald, Editors,
Crystallographic groups and their generalizations, 2000 261 Gregory Budzban, Philip Feinsilver, and Arun Mukherjea, Editors, Probability
on algebraic structures, 2000 260 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, First summer school in
analysis and mathematical physics: Quantization, the Segal-Bargmann transform and semiclassical analysis, 2000
259 D. V. Huynh, S. K. Jain, and S. R. L6pez-Permouth, Editors, Algebra and its applications, 2000
258 Karsten Grove, Ib Henning Madsen, and Erik Kjrer Pedersen, Editors, Geometry and topology: Aarhus, 2000
257 Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, Editors, Computability theory and its applications: Current trends and open problems, 2000
256 Irwin Kra and Bernard Maskit, Editors, In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, 2000
255 Jerry Bona, Katarzyna Saxton, and Ralph Saxton, Editors, Nonlinear PDE's, dynamics and continuum physics, 2000
254 Mourad E. H. Ismail and Dennis W. Stanton, Editors, q-series from a contemporary perspective, 2000
253 Charles N. Delzell and James J. Madden, Editors, Real algebraic geometry and ordered structures, 2000
252 Nathaniel Dean, Cassandra M. McZeal, and Pamela J. Williams, Editors, African Americans in Mathematics II, 1999
251 Eric L. Grinberg, Shiferaw Berhanu, Marvin I. Knopp, Gerardo A. Mendoza, and Eric Todd Quinto, Editors, Analysis, geometry, number theory: The Mathematics of Leon Ehrenpreis, 2000
250 Robert H. Gilman, Editor, Groups, languages and geometry, 1999 249 Myung-Hwan Kim, JohnS. Hsia, Yoshiyuki Kitaoka, and Rainer
Schulze-Pillot, Editors, Integral quadratic forms and lattices, 1999 (Continued in the back of this publication)
http://dx.doi.org/10.1090/conm/272
Quadratic Forms and Their Applications
CoNTEMPORARY MATHEMATICS
272
Quadratic Forms and Their Applications
Proceedings of the Conference on Quadratic Forms and Their Applications
July 5-9, 1999 University College Dublin
Eva Bayer-Fiuckiger, David Lewis, Andrew Ranicki, Editors
American Mathematical Society Providence, Rhode Island
Editorial Board Dennis DeThrck, managing editor
Andreas Blass Andy R. Magid Michael Vogelius This volume contains the proceedings of the conference on "Quadratic Forms and Their
Applications", which was held at University College Dublin July 5-9, 1999. Some of the papers are expanded versions of the one-hour survey lectures delivered at the conference. Others are devoted to current research and are based on the half-hour lectures, while others are concerned with the history of quadratic forms.
The conference was supported by the European Community under the auspices of the TMR network FMRX CT-97-0107 "Algebraic K-Theory, Linear Algebraic Groups and Related Structures" .
2000 Mathematics Subject Classification. Primary llExx, OlAxx.
Library of Congress Cataloging-in-Publication Data Conference on Quadratic Forms and Their Applications (1999 : University College Dublin)
Quadratic forms and their applications : proceedings of the Conference on Quadratic Forms and Their Applications, July 5-9, 1999, University College Dublin / Eva Bayer-Fluckiger, David Lewis, Andrew Ranicki, editors.
p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 272) Includes bibliographical references. ISBN 0-8218-2779-0 (alk. paper) 1. Forms, Quadratic-Congresses.
III. Ranicki, Andrew, 1948- IV. Title. Society) ; v. 272.
I. Bayer-Fluckiger, Eva, 1951-- II. Lewis, David. V. Contemporary mathematics (American Mathematical
QA243 .C753 1999 5121.7 4-dc21 00-053584
Copying and reprinting. Material in this book may be reproduced by any means for edu-cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg-ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permission@ams. or g.
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§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
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Contents
Preface 1x
Conference lectures x
Conference participants xii
Conference photo x1v
Galois cohomology of the classical groups EvA BAYER-FLUCKIGER 1
Symplectic lattices ANNE-MARIE BERGE 9
Universal quadratic forms and the Fifteen Theorem J. H. CONWAY 23
On the Conway-Schneeberger Fifteen Theorem MANJUL BHARGAVA 27
On trace forms and the Burnside ring MARTIN EPKENHANS 39
Equivariant Brauer groups A. FROHLICH AND C.T.C. WALL 57
Isotropy of quadratic forms and field invariants DETLEV W. HOFFMANN 73
Quadratic forms with absolutely maximal splitting 0LEG IZHBOLDIN AND ALEXANDER VISHIK 103
2-regularity and reversibility of quadratic mappings ALEXEY F. IZMAILOV 127
Quadratic forms in knot theory C. KEARTON 135
Biography of Ernst Witt (1911-1991) INA KERSTEN 155
Generic splitting towers and generic splitting preparation of quadratic forms MANFRED KNEBUSCH AND ULF REHMANN 173
vii
viii CONTENTS
Local densities of hermitian forms MAURICE MISCHLER 201
Notes towards a constructive proof of Hilbert's theorem on ternary quartics VICTORIA POWERS AND BRUCE REZNICK 209
On the history of the algebraic theory of quadratic forms WINFRIED SCHARLAU 229
Local fundamental classes derived from higher K -groups: III VICTOR P. SNAITH 261
Hilbert's theorem on positive ternary quartics RICHARD G. SWAN 287
Quadratic forms and normal surface singularities C.T.C. WALL 293
Preface
These are the proceedings of the conference on "Quadratic Forms and Their Applications" which was held at University College Dublin from 5th to 9th July, 1999. The meeting was attended by 82 participants from Europe and elsewhere. There were 13 one-hour lectures surveying various applications of quadratic forms in algebra, number theory, algebraic geometry, topology and information theory. In addition, there were 22 half-hour lectures on more specialized topics.
The papers collected together in these proceedings are of various types. Some are expanded versions of the one-hour survey lectures delivered at the conference. Others are devoted to current research and are based on the half-hour lectures. Yet others are concerned with the history of quadratic forms. All papers were refereed, and we are grateful to the referees for their work.
This volume includes one of the last papers of Oleg Izhboldin, who died unex-pectedly on 17th April 2000 at the age of 37. His untimely death is a great loss to mathematics and in particular to quadratic form theory. We shall miss his brilliant and original ideas, his clarity of exposition, and his friendly and good-humoured presence.
The conference was supported by the European Community under the auspices of the TMR network FMRX CT-97-0107 "Algebraic K-Theory, Linear Algebraic Groups and Related Structures". We are grateful to the Mathematics Department of University College Dublin for hosting the conference, and in particular to Thomas Unger for all his work on the TEX and web-related aspects of the conference.
ix
Eva Bayer-Fluckiger, Besan~on David Lewis, Dublin Andrew Ranicki, Edinburgh October, 2000
Conference lectures
60 minutes. A.-M. BERGE, Symplectic lattices. J.J. BoUTROS, Quadratic forms in information theory. J.H. CONWAY, The Fifteen Theorem. D. HOFFMANN, Zeros of quadratic forms. C. KEARTON, Quadratic forms in knot theory. M. KRECK, Manifolds and quadratic forms. R. PARIMALA, Algebras with involution. A. PFISTER, The history of the Milnor conjectures. M. RosT, On characteristic numbers and norm varieties. W. SCHARLAU, The history of the algebraic theory of quadratic .forms. J.-P. SERRE, Abelian varieties and hermitian modules. M. TAYLOR, Galois modules and hermitian Euler characteristics. C.T.C. WALL, Quadratic forms in singularity theory.
30 minutes. A. ARUTYUNOV, Quadratic .forms and abnormal extremal problems: some results and unsolved problems. P. BALMER, The Witt groups of triangulated categories, with some applications. G. BERHUY, Hermitian scaled trace forms of field extensions. P. CALAME, Integral forms without symmetry. P. CHUARD-KOULMANN, Elements of given minimal polynomial in a central simple algebra. M. EPKENHANS, On trace forms and the Burnside ring. L. FAINSILBER, Quadratic .forms and gas dynamics: sums of squares in a discrete velocity model for the Boltzmann equation. C. FRINGS, Second trace form and T2 -standard normal bases. J. HURRELBRINK, Quadratic forms of height 2 and differences of two Pfister .forms. M. IFTIME, On spacetime distributions. A. IZMAILOV, 2-regularity and reversibility of quadratic mappings. S. JOUKHOVITSKI, K-theory of the Weil transfer .functor. V. MAUDUIT, Towards a Drin.feldian analogue of quadratic forms .for polynomials. M. MISCHLER, Local densities and Jordan decomposition. V. POWERS, Computational approaches to Hilbert's theorem on ternary quartics. S. PUMPLlJN, The Witt ring of a Brauer-Severi variety. A. QUEGUINER, Discriminant and Clifford algebras of an algebra with involution. U. REHMANN, A surprising fact about the generic splitting tower of a quadratic .form.
X
CONFERENCE LECTURES xi
C. RIEHM, Orthogonal representations of finite groups. D. SHEIHAM, Signatures of Seifert forms and cobordism of boundary links. V. SNAITH, Local fundamental classes constructed from higher dimensional K-groups. K. ZAINOULLINE, On Grothendieck 's conjecture about principal homogeneous spaces.
Conference participants
A.V. Arutyunov, Moscow P. Balmer, Lausanne E. Bayer-Fluckiger, Besan<;on K.J. Becher, Besan<;on A.-M. Berge, Talence G. Berhuy, Besan<;on J.J. Boutros, Paris L. Broecker, Muenster Ph. Calame, Lausanne P. Chuard-Koulmann, Louvain-la-N euve J.H. Conway, Princeton J .-Y. Degos, Talence S. Dineen, Dublin Ph. Du Bois, Angers G. Elencwajg, Nice M. Elhamdadi, Trieste M. Elomary, Louvain-la-Neuve M. Epkenhans, Paderborn L. Fainsilber, Goteborg D.A. Flannery, Cork C. Frings, Besan<;on R.S. Garibaldi, Zurich S. Gille, Muenster :M. Gindraux, Neuchatel R. Cow. Dublin F. Gradl, Duisburg Y. Hellegouarch, Caen D. Hoffmann, Besan<;on J. Hurrelbrink, Baton Rouge K. Hutchinson. Dublin M. lftime, Suceava A.F. Izmailov, Moscow S. Joukhovitski, Bonn M. Karoubi, Paris C. Kearton, Durham M. Kreck, Heidelberg T .J. Laffey, Dublin
xii
arutunov(Q)sa640.cs.msu.su [email protected] bayer@vega. univ-fcomte.fr becher@math. univ-fcomte.fr berge@math. u-bordeaux.fr berhuy@math. univ-fcomte.fr [email protected] broe@math. uni-muenster .de philippe.calame@ima. unil.ch [email protected] conway@math. princeton.edu [email protected] sean.dineenCQiucd.ie [email protected] elenc@math. unice.fr melha@ictp. trieste .it elomary@agel. ucl.ac. be [email protected] [email protected] dfiannery@cit .ie fringslQlmath. univ-fcomte.fr [email protected] gilles@math. uni-muenster .de marc.gindraux@maths. unine.ch [email protected] gradlCQlmath. uni-duisburg.de hellegou @math. unicaen.fr [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]. uk kreck@mathi. uni-heidelberg.de thomas.laffeylQlucd.ie
C'ONFEHENCE PARTICIPANTS
C. Lamy, Paris A. Leibak, Tallinn D. Lewis, Dublin J.-L. Loday, Strasbourg M. Mackey, Dublin M. Marjoram, Dublin V. Mauduit, Dublin A. Mazzoleni, Lausanne S. McGarraghy, Dublin G. McGuire, Maynooth M. Mischler, Lausanne M. Monsurro, Besan<;on J. Morales, Baton Rouge C. Mulcahy, Atlanta H. Munkholm, Odense R. Parimala, Besan<;on 0. Patashnick, Chicago S. Perret, Neuchatel A. Pfister, Mainz V. Powers, Atlanta S. Pumplun, Regensburg A. Queguiner, Paris A. Ranicki, Edinburgh U. Rehmann, Bielefeld H. Reich, Muenster C. Riehm, Hamilton, Ontario M. Rost, Regensburg
D. Ryan, Dublin W. Scharlau, Muenster C. Scheiderer, Duisburg J.-P. Serre, Paris D. Sheiham, Edinburgh F. Sigrist, Neuchatel V. Snaith, Southampton M. Taylor, Manchester J.-P. Tignol, Louvain-la-Neuve U. Tipp, Ghent D.A. Tipple, Dublin M. Tuite, Galway T. Unger, Dublin C.T.C. Wall, Liverpool S. Yagunov, London Ontario K. Zahidi, Ghent K. Zainoulline, St. Petersburg
[email protected] [email protected] [email protected] loday@math. u-strasbg.fr [email protected] [email protected] [email protected] amedeo.mazzoleni@ima. unil.ch [email protected] [email protected] maurice.mischler@ima. unil.ch monsurro@math. univ-fcomte.fr [email protected] [email protected] [email protected] parimala@vega. univ-fcomte.fr owen@math. uchicago.edu stephan e. perret@maths. unine.ch pfister@mathematik. uni-mainz.de [email protected] [email protected]. uni tn.it [email protected] [email protected]. uk [email protected] reichh@escher. uni-muenster .de [email protected] markus.rost@mathematik.
uni-regensburg.de dermot. w [email protected] [email protected] claus@math. uni-duisburg.de [email protected] [email protected]. uk francois.sigrist@maths. unine.ch [email protected]. uk [email protected]. uk tignol@agel. ucl.ac. be [email protected] david. [email protected] michael. [email protected] thomas. [email protected] ctcw@li verpool. ac. uk [email protected]. uwo.ca [email protected] [email protected]
xiii
(1)
A.
Ran
icki
, (2
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H.
Con
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. E
lenc
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(5)
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(7)
S. P
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(8
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. B
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(9)
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(10)
M.
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11)
M.
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surr
o, (
12)
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, (1
3) E
. B
ayer
-Flu
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(14)
D.
Lew
is,
(15)
R.
Pari
mal
a, (
16)
V.
Pow
-er
s,
(17)
F.
Sigr
ist,
(18)
H
. R
eich
, (1
9)
K.
Zahi
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(20)
C.T
.C.
Wal
l, (2
1)
H.
Mun
khol
m,
(22)
M
. If
tim
e, (
23)
A.
Lei
bak,
(24
) C
. Fr
ings
, (2
5) C
. R
iehm
, (2
6) M
. M
isch
ler,
(27
) J.
-P.
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e, (
28)
S. P
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t, (
29)
A.-
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ge,
(30)
W.
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, (3
1) M
. K
reck
, (3
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. G
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(33
) J.
-Y
. D
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, (3
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.S.
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(35)
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arra
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(36
) 0
. Pa
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, (3
7) P
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(38
) G
. M
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(39)
C.
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, (4
0) D
. Sh
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(41)
Ph.
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(42
) L.
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(43
) J.
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(44)
J.-
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(45)
M.
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(46
) J.
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, (4
7) A
. M
azzo
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, (4
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. Sc
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, (4
9) A
.F. I
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lov,
(50
) K
. Hut
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(51
) A
.V. A
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, (5
2) J
. Hur
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, (53
) A
. Pfi
ster
, (5
4) M
. Ros
t, (5
5) Y
. Hel
lego
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56)
T. U
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, (5
7) P
. C
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. H
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) M
. G
indr
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. E
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, (6
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.J.
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(63)
K.J
. B
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64)
M.
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, (6
5) F
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, (6
6) P
. B
alm
er,
(67)
D.
Flan
nery
, (6
8) M
. T
uite
.
Selected Titles in This Series (Continued from the front of this publication)
248 Naihuan Jing and Kailash C. Misra, Editors, Recent developments in quantum affine algebras and related topics, 1999
247 Lawrence Wasson Baggett and David Royal Larson, Editors, The functional and harmonic analysis of wavelets and frames, 1999
246 Marcy Barge and Krystyna Kuperberg, Editors, Geometry and topology in dynamics, 1999
24.5 Michael D. Fried, Editor, Applications of curves over finite fields, 1999 244 Leovigildo Alonso Tarrio, Ana Jeremias Lopez, and Joseph Lipman, Studies in
duality on noetherian formal schemes and non-noetherian ordinary schemes, 1999 243 Tsit Yuan Lam and Andy R. Magid, Editors, Algebra, K-theory, groups, and
education, 1999 242 Bernheim Booss-Bavnbek and Krzysztof Wojciechowski, Editors, Geometric
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in discrete and computational geometry, 1999
For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej.
ISBN 0-8218-2779-0
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