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 Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berg ´ e 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. Fr ¨ ohlich and C.T.C. Wall 57 Isotropy of quadratic forms and field invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg 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 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 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 ix Preface These are the pro ceedings 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 appli- cations of quadratic forms in algebra, number theory, algebraic geometry, topology and information theory. In addition, there were 22 half-hour lec- tures 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 unexpectedly 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 con- ference, and in particular to Thomas Unger for all his work on the T E X and web-related aspects of the conference. Eva Bayer-Fluckiger, Besan¸con David Lewis, Dublin Andrew Ranicki, Edinburgh October, 2000 x Conference lectures 60 minutes. A M. Berg ´ e, 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 applica- tions. 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 T 2 -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. xi V. Mauduit, Towards a Drinfeldian analogue of quadratic forms for poly- nomials. M. Mischler, Local densities and Jordan decomposition. V. Powers, Computational approaches to Hilbert’s theorem on ternary quartics. S. Pumpl ¨ un, The Witt ring of a Brauer-Severi variety. A. Qu ´ eguiner, Discriminant and Clifford algebras of an algebra with in- volution. U. Rehmann, A surprising fact about the generic splitting tower of a qua- dratic form. 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 homoge- neous spaces. xii Conference participants A.V. Arutyunov, Moscow arutunov@sa640.cs.msu.su P. Balmer, Lausanne balmer@math.rutgers.edu E. Bayer-Fluckiger, Besan¸con bayer@vega.univ-fcomte.fr K.J. Becher, Besan¸con becher@math.univ-fcomte.fr A M. Berg´e, Talence berge@math.u-bordeaux.fr G. Berhuy, Besan¸con berhuy@math.univ-fcomte.fr J.J. Boutros, Paris boutros@com.enst.fr L. Broecker, Muenster broe@math.uni-muenster.de Ph. Calame, Lausanne philippe.calame@ima.unil.ch P. Chuard-Koulmann, Louvain-la-Neuve chuard@agel.ucl.ac.be J.H. Conway, Princeton conway@math.princeton.edu J Y. Degos, Talence degos@math.u-bordeaux.fr S. Dineen, Dublin sean.dineen@ucd.ie Ph. Du Bois, Angers pdubois@univ-angers.fr G. Elencwajg, Nice elenc@math.unice.fr M. Elhamdadi, Trieste melha@ictp.trieste.it M. Elomary, Louvain-la-Neuve elomary@agel.ucl.ac.be M. Epkenhans, Paderborn martine@uni-paderborn.de L. Fainsilber, Goteborg laura@math.chalmers.se D.A. Flannery, Cork dflannery@cit.ie C. Frings, Besan¸con frings@math.univ-fcomte.fr R.S. Garibaldi, Zurich skip@math.ethz.ch S. Gille, Muenster gilles@math.uni-muenster.de M. Gindraux, Neuchatel marc.gindraux@maths.unine.ch R. Gow, Dublin rod.gow@ucd.ie F. Gradl, Duisburg gradl@math.uni-duisburg.de Y. Hellegouarch, Caen hellegou@math.unicaen.fr D. Hoffmann, Besan¸con detlev@vega.univ-fcomte.fr J. Hurrelbrink, Baton Rouge jurgen@julia.math.lsu.edu K. Hutchinson, Dublin kevin.hutchinson@ucd.ie M. Iftime, Suceava mtime@warpnet.ro A.F. Izmailov, Moscow izmaf@ccas.ru S. Joukhovitski, Bonn seva@math.nwu.edu M. Karoubi, Paris karoubi@math.jussieu.fr C. Kearton, Durham cherry.kearton@durham.ac.uk M. Kreck, Heidelberg kreck@mathi.uni-heidelberg.de T.J. Laffey, Dublin thomas.laffey@ucd.ie C. Lamy, Paris lamyc@enst.fr A. Leibak, Tallinn aleibak@ioc.ee D. Lewis, Dublin david.lewis@ucd.ie J L. Loday, Strasbourg loday@math.u-strasbg.fr M. Mackey, Dublin michael.mackey@ucd.ie xiii M. Marjoram, Dublin martin.marjoram@ucd.ie V. Mauduit, Dublin veronique.mauduit@ucd.ie A. Mazzoleni, Lausanne amedeo.mazzoleni@ima.unil.ch S. McGarraghy, Dublin john.mcgarraghy@ucd.ie G. McGuire, Maynooth gmg@maths.may.ie M. Mischler, Lausanne maurice.mischler@ima.unil.ch M. Monsurro, Besan¸con monsurro@math.univ-fcomte.fr J. Morales, Baton Rouge morales@math.lsu.edu C. Mulcahy, Atlanta colm@spelman.edu H. Munkholm, Odense hjm@imada.sdu.dk R. Parimala, Besan¸con parimala@vega.univ-fcomte.fr O. Patashnick, Chicago owen@math.uchicago.edu S. Perret, Neuchatel stephane.perret@maths.unine.ch A. Pfister, Mainz pfister@mathematik.uni-mainz.de V. Powers, Atlanta vicki@mathcs.emory.edu S. Pumpl¨un, Regensburg pumplun@degiorgi.science.unitn.it A. Qu´eguiner, Paris queguin@math.univ-paris13.fr A. Ranicki, Edinburgh aar@maths.ed.ac.uk U. Rehmann, Bielefeld rehmann@mathematik.uni-bielefeld.de H. Reich, Muenster reichh@escher.uni-muenster.de C. Riehm, Hamilton, Ontario riehm@mcmail.cis.mcmaster.ca M. Rost, Regensburg markus.rost@mathematik.uni-regensburg.de D. Ryan, Dublin dermot.w.ryan@ucd.ie W. Scharlau, Muenster scharlau@escher.uni-muenster.de C. Scheiderer, Duisburg claus@math.uni-duisburg.de J P. Serre, Paris jean-pierre.serre@ens.fr D. Sheiham, Edinburgh des@maths.ed.ac.uk F. Sigrist, Neuchatel francois.sigrist@maths.unine.ch V. Snaith, Southampton vps@maths.soton.ac.uk M. Taylor, Manchester mcbtayr@mail1.mcc.ac.uk J P. Tignol, Louvain-la-Neuve tignol@agel.ucl.ac.be U. Tipp, Ghent utipp@cage.rug.ac.be D.A. Tipple, Dublin david.tipple@ucd.ie M. Tuite, Galway michael.tuite@nuigalway.ie T. Unger, Dublin thomas.unger@ucd.ie C.T.C. Wall, Liverpool ctcw@liverpool.ac.uk S. Yagunov, London Ontario yagunov@jardine.math.uwo.ca K. Zahidi, Ghent karim.zahidi@rug.ac.be K. Zainoulline, St. Petersburg kirill@pdmi.ras.ru xiv Conference photo xv 1 3 2 4 5 7 8 10 11 13 12 14 15 16 17 18 19 6 20 9 28 29 30 44 45 57 43 55 56 58 22 23 21 32 31 33 34 35 59 46 61 60 62 47 48 63 64 36 24 37 49 65 50 66 38 39 25 26 40 52 51 67 68 53 54 41 27 42 (1) A. Ranicki, (2) J.H. Conway, (3) G. Elencwa jg, (4) K. Zainoulline, (5) M. Elomary, (6) V. Snaith, (7) S. Pumpl¨un, (8) G. Berhuy, (9) A. Qu´equiner, (10) M. Du Bois, (11) M. Monsurro, (12) V. Mauduit, (13) E. Bayer-Fluckiger, (14) D. Lewis, (15) R. Parimala, (16) V. Pow- ers, (17) F. Sigrist, (18) H. Reich, (19) K. Zahidi, (20) C.T.C. Wall, (21) H. Munkholm, (22) M. Iftime, (23) A. Leibak, (24) C. Frings, (25) C. Riehm, (26) M. Mischler, (27) J P. Serre, (28) S. Perret, (29) A M. Berg´e, (30) W. Scharlau, (31) M. Kreck, (32) S. Gille, (33) J Y. Degos, (34) R.S. Garibaldi, (35) S. McGarraghy, (36) O. Patashnick, (37) Ph. Du Bois, (38) G. McGuire, (39) C. Mulcahy, (40) D. Sheiham, (41) Ph. Calame, (42) L. Fainsilber, (43) J P. Tignol, (44) J L. Loday, (45) M. Taylor, (46) J. Morales, (47) A. Mazzoleni, (48) C. Scheiderer, (49) A.F. Izmailov, (50) K. Hutchinson, (51) A.V. Arutyunov, (52) J. Hurrelbrink, (53) A. Pfister, (54) M. Rost, (55) Y. Hellegouarch, (56) T. Unger, (57) P. Chuard-Koulmann, (58) D. Hoffmann, (59) M. Gindraux, (60) U. Tipp, (61) M. Epkenhans, (62) T.J. Laffey, (63) K.J. Becher, (64) M. El- hamdadi, (65) F. Gradl, (66) P. Balmer, (67) D. Flannery, (68) M. Tuite. [...]... the 15-theorem, and derives the complete list of universal quaternaries As he remarks, of the 204 such forms, Willerding’s purportedly complete list of 178 contains in fact only 168, because she missed 36 forms, listed 1 form twice, and listed 9 nonuniversal forms! UNIVERSAL QUADRATIC FORMS AND THE FIFTEEN THEOREM 25 The 15-theorem closes the universality problem for integer-matrix forms by providing... theorem” for quadratic forms [26] : if q1 , q2 and q are quadratic forms such that q1 ⊕ q q2 ⊕ q, then q1 q2 The analog of this result for hermitian forms over skew fields also holds, see for instance [8] or [15] These results can also be deduced from a statement on linear algebraic groups due to Borel and Tits : Theorem 2.3.1 ([20], III.2.1., Exercice 1) Let G be a connected reductive group, and P a parabolic... (k, G) is injective 3 Classification of quadratic forms and Galois cohomology Recall (cf 1.5.) that if Oq is the orthogonal group of a non–degenerate, n– dimensional quadratic form q over k, then H 1 (k, Oq ) is the set of isomorphism classes of non–degenerate quadratic forms over k of dimension n Hence determining this set is equivalent to classifying quadratic forms over k up to isomorphism The cohomological... finite field It is well–known that non–degenerate quadratic forms over k are determined by their dimension and discriminant Hence by 3.1 we have H 1 (k, SOq ) = 0 for all q We can go one step further, and consider an H 2 –invariant (the Hasse–Witt invariant) that will suffice, together with dimension and discriminant, to classify non–degenerate quadratic forms over certain fields Let Spinq be the spin group... berge@math.u-bordeaux.fr Contemporary Mathematics Universal Quadratic Forms and the Fifteen Theorem J H Conway Abstract This paper is an extended foreword to the paper of Manjul Bhargava [1] in these proceedings, which gives a short and elegant proof of the Conway-Schneeberger Fifteen Theorem on the representation of integers by quadratic forms The representation theory of quadratic forms has a long history, starting in... Z/2Z) Recall that we have H 1 (k) k ∗ /k ∗2 and H 2 (k) Br2 (k) If q < a1 , , an > is a non–degenerate quadratic form, we define the discrimn(n−1) inant of q by disc(q) = (−1) 2 a1 an ∈ k ∗ /k ∗2 , and the Hasse–Witt invariant by w2 (q) = Σi . 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. 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 quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the pro ceedings of the conference on Quadratic Forms And Their Applications which