Differential Forms, Linked Fields and the $ U $-Invariant
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Minimal Weakly Isotropic Forms
Minimal weakly isotropic forms Karim Johannes Becher Abstract In this article weakly isotropic quadratic forms over a (formally) real field are studied. Conditions on the field are given which imply that every weakly isotropic form over that field has a weakly isotropic sub- form of small dimension. Fields over which every quadratic form can be decomposed into an orthogonal sum of a strongly anisotropic form and a torsion form are characterized in different ways. Keywords: quadratic form, weakly isotropic, strongly anisotropic, real field, Pythagoras number, field invariants, SAP-field, local-global principle Classification (MSC 2000): 11E04, 11E81, 12D15 1 Introduction The concept of weak isotropy was introduced by Prestel in [11] and studied in the 1970ies, especially by Prestel and Br¨ocker. The main motivation for this concept is the failure of any kind of local-global principle for isotropy of quadratic forms in a general situation. It turned out that, over any (formally) real field, weakly isotropic forms can be characterized, on the one hand in terms of the semi-orderings (cf. [11]), on the other hand in terms of the real places of the field (cf. [11], [2]). Applications of weakly isotropic forms occur in real algebra and geometry (cf. [13]). The present article investigates weakly isotropic forms in view of the question what kind of subforms and orthogonal decompositions they admit. This leads to consider in particular those weakly isotropic forms which are minimal in the sense that they do not contain any proper subform which is weakly isotropic. The question of how large the dimension of such a form may be gives rise to a new field invariant. -
Garibaldi S. Cohomological Invariants.. Exceptional Groups And
Cohomological Invariants: Exceptional Groups and Spin Groups This page intentionally left blank EMOIRS M of the American Mathematical Society Number 937 Cohomological Invariants: Exceptional Groups and Spin Groups Skip Garibaldi ÜÌ Ê>Ê««i`ÝÊLÞÊ iÌiÛÊ7°Êvv>® ÕÞÊÓääÊÊUÊÊ6ÕiÊÓääÊÊUÊÊ ÕLiÀÊÎÇÊÃiV`ÊvÊÈÊÕLiÀîÊÊUÊÊ-- ÊääÈxÓÈÈ American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 11E72; Secondary 12G05, 20G15, 17B25. Library of Congress Cataloging-in-Publication Data Garibaldi, Skip, 1972– Cohomological invariants : exceptional groups and spin groups / Skip Garibaldi ; with an appendix by Detlev W. Hoffmann. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 937) “Volume 200, number 937 (second of 6 numbers).” Includes bibliographical references and index. ISBN 978-0-8218-4404-5 (alk. paper) 1. Galois cohomology. 2. Linear algebraic groups. 3. Invariants. I. Title. QA169.G367 2009 515.23—dc22 2009008059 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. The 2009 subscription begins with volume 197 and consists of six mailings, each containing one or more numbers. Subscription prices for 2009 are US$709 list, US$567 institutional member. A late charge of 10% of the subscription price will be imposed on orders received from nonmembers after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of US$65; subscribers in India must pay a postage surcharge of US$95. Expedited delivery to destinations in North America US$57; elsewhere US$160. Each number may be ordered separately; please specify number when ordering an individual number. -
Historical Background
Chapter 0 Historical Background The theory of composition of quadratic forms over fields had its start in the 19th century with the search for n-square identities of the type 2 + 2 +···+ 2 · 2 + 2 +···+ 2 = 2 + 2 +···+ 2 (x1 x2 xn) (y1 y2 yn) z1 z2 zn where X = (x1,x2,...,xn) and Y = (y1,y2,...,yn) are systems of indeterminates and each zk = zk(X, Y ) is a bilinear form in X and Y . For example when n = 2 there is the ancient identity 2 + 2 · 2 + 2 = + 2 + − 2 (x1 x2 ) (y1 y2 ) (x1y1 x2y2) (x1y2 x2y1) . In this example z1 = x1y1 + x2y2 and z2 = x1y2 − x2y1 are bilinear forms in X, Y with integer coefficients. This formula for n = 2 can be interpreted as the “law of moduli” for complex numbers: |α|·|β|=|αβ| where α = x1 −ix2 and β = y1 +iy2. A similar 4-square identity was found by Euler (1748) in his attempt to prove Fermat’s conjecture that every positive integer is a sum of four integer squares. This identity is often attributed to Lagrange, who used it (1770) in his proof of that conjec- ture of Fermat. Here is Euler’s formula, in our notation: 2 + 2 + 2 + 2 · 2 + 2 + 2 + 2 = 2 + 2 + 2 + 2 (x1 x2 x3 x4 ) (y1 y2 y3 y4 ) z1 z2 z3 z4 where z1 = x1y1 + x2y2 + x3y3 + x4y4 z2 = x1y2 − x2y1 + x3y4 − x4y3 z3 = x1y3 − x2y4 − x3y1 + x4y2 z4 = x1y4 + x2y3 − x3y2 − x4y1. After Hamilton’s discovery of the quaternions (1843) this 4-square formula was interpreted as the law of moduli for quaternions. -
Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Non-Excellence
GENERIC SPLITTING FIELDS OF CENTRAL SIMPLE ALGEBRAS: GALOIS COHOMOLOGY AND NON-EXCELLENCE OLEG T. IZHBOLDIN AND NIKITA A. KARPENKO Abstract. A field extension L=F is called excellent, if for any quadratic form ϕ over F the anisotropic part (ϕL)an of ϕ over L is defined over F ; L=F is called universally excellent, if L · E=E is excellent for any field ex- tension E=F . We study the excellence property for a generic splitting field of a central simple F -algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi-Brauer varieties and its relationship with the relative Galois co- homology group H3(L=F ) for a generic (common) splitting field L of the corresponding central simple F -algebras. Let F be a field and let A1;:::;An be central simple F -algebras. A splitting field of this collection of algebras is a field extension L of F such that all the def algebras (Ai)L = Ai ⊗F L (i = 1; : : : ; n) are split. A splitting field L=F of 0 A1;:::;An is called generic (cf. [41, Def. C9]), if for any splitting field L =F 0 of A1;:::;An, there exists an F -place of L to L . Clearly, generic splitting field of A1;:::;An is not unique, however it is uniquely defined up to equivalence over F in the sense of [24, x3]. Therefore, working on problems (Q1) and (Q2) discussed below, we may choose any par- ticular generic splitting field L of the algebras A1;:::;An (cf. -
Isotropy of Quadratic Spaces in Finite and Infinite Dimension
Documenta Math. 473 Isotropy of Quadratic Spaces in Finite and Infinite Dimension Karim Johannes Becher, Detlev W. Hoffmann Received: October 6, 2006 Revised: September 13, 2007 Communicated by Alexander Merkurjev Abstract. In the late 1970s, Herbert Gross asked whether there exist fields admitting anisotropic quadratic spaces of arbitrarily large finite dimensions but none of infinite dimension. We construct exam- ples of such fields and also discuss related problems in the theory of central simple algebras and in Milnor K-theory. 2000 Mathematics Subject Classification: 11E04, 11E81, 12D15, 12E15, 12F20, 12G05, 12G10, 16K20, 19D45 Keywords and Phrases: quadratic form, isotropy, infinite-dimensional quadratic space, u-invariant, function field of a quadric, totally indef- inite form, real field, division algebra, quaternion algebra, symbol al- gebra, Galois cohomology, cohomological dimension, Milnor K-theory 1 Introduction A quadratic space over a field F is a pair (V, q) of a vector space V over F together with a map q : V F such that −→ q(λx) = λ2q(x) for all λ F , x V , and • ∈ ∈ the map bq : V V F defined by bq(x,y) = q(x + y) q(x) q(y) • (x,y V ) is F -bilinear.× −→ − − ∈ If dim V = n < , one may identify (after fixing a basis of V ) the quadratic space (V, q) with∞ a form (a homogeneous polynomial) of degree 2 in n variables. Via this identification, a finite-dimensional quadratic space over F will also be referred to as quadratic form over F . Recall that a quadratic space (V, q) is Documenta Mathematica 12 (2007) 473–504 474 Karim Johannes Becher, Detlev W. -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Orthogonal Symmetries and Clifford Algebras 11
ORTHOGONAL SYMMETRIES AND CLIFFORD ALGEBRAS M. G. MAHMOUDI Abstract. Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated. 2000 Mathematics Subject Classification: 16W10, 11E39 Key words: Orthogonal symmetry, reflection, Clifford algebra, Involution, qua- dratic form, even Clifford algebra, multiquaternion algebra, universal property of Clifford algebra, Clifford map 1. Introduction Clifford algebra is one of the important algebraic structures which can be associ- ated to a quadratic form. These algebras are among the most fascinating algebraic structures. Not only they have many applications in algebra and other branches of Mathematics, but also they have wide applications beyond Mathematics, e.g., Physics, Computer Science and Engineering [6], [18], [14], [15]. A detailed historical account of Clifford algebras from their genesis can be found in [21]. See also [7] and [14] for an interesting brief historical account of Clifford algebras. Many familiar algebras can be regarded as special cases of Clifford algebras. For example the algebra of Complex numbers is isomorphic to the Clifford algebra of any one-dimensional negative definite quadratic form over R. The algebra of Hamilton quaternions is isomorphic to the Clifford algebras of a two-dimensional negative definite quadratic form over R. More generally, as shown by D. Lewis in [13, Proposition 1], a multiquaternion algebra, i.e., an algebra like Q1 Qn ⊗···⊗ff where Qi is a quaternion algebra over a field K, can be regarded as the Cli ord algebra of a suitable nondegenerate quadratic form q over the base field K. In [13], such a form q is also explicitly constructed. -
SMALL REPRESENTATIONS of INTEGERS by INTEGRAL QUADRATIC FORMS 1. Introduction and Statement of Results the Question of Effective
SMALL REPRESENTATIONS OF INTEGERS BY INTEGRAL QUADRATIC FORMS WAI KIU CHAN AND LENNY FUKSHANSKY Abstract. Given an isotropic quadratic form over a number field which as- sumes a value t, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of quadratic forms, we produce bounds on height of points outside of some algebraic sets in a quadratic space at which the form assumes the value t. Our bounds on height are explicit in terms of the heights of the form, the space, the algebraic set and the value t. 1. Introduction and statement of results The question of effective distribution of zeros of quadratic forms has a long history. In particular, the famous Cassels' theorem (1955) asserts that an isotropic rational quadratic form has zeros of bounded height with effective bounds depending on the height of this form, [2]. By height here we mean a height function, a standard tool of Diophantine geometry, measuring the arithmetic complexity of an object; we provide a brief overview of height functions in Section 2. There have been many extensions and generalizations of Cassels theorem over the years, see [7] for a survey of this lively area. In particular, in the recent years there have been several results assessing the distribution of small-height zeros by proving that they cannot be easily \cut-out" by any finite union of linear subspaces or even projective varieties not completely containing the quadratic hypersurface in question. Specifically, results of [6], [5], [3], [8] prove existence of zeros of bounded height of an isotropic quadratic form outside of a union of projective varieties. -
Classification of Quadratic Forms with Clifford Algebras
CLASSIFICATION OF QUADRATIC FORMS WITH CLIFFORD ALGEBRAS Dorothy Cheung Supervisor: Associate Professor Daniel Chan School of Mathematics and Statistics The University of New South Wales October 2015 Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science with Honours Plagiarism statement I declare that this thesis is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere. I acknowledge that the assessor of this thesis may, for the purpose of assessing it: • Reproduce it and provide a copy to another member of the University; and/or, • Communicate a copy of it to a plagiarism checking service (which may then retain a copy of it on its database for the purpose of future plagiarism check- ing). I certify that I have read and understood the University Rules in respect of Student Academic Misconduct, and am aware of any potential plagiarism penalties which may apply. By signing this declaration I am agreeing to the statements and conditions above. Signed: Date: i Acknowledgements First and foremost, my greatest gratitude goes to my supervisor, Associate Professor Daniel Chan, who has provided much needed guidance throughout the year. He was always more than willing to answer my questions, patiently explaining things on the whiteboard and in general gave me invaluable advice on how to best tackle my Honours years. Also, I want to give thanks to Professor Jie Du, who was my supervisor for vacation research project. He offered great advice and is always passionate about mathemat- ics. I also want to thank Associate Professor Catherine Greenhill for her insightful feedbacks during the Honours talks. -
Quadratic Forms and Their Applications
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 Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld 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 proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999. -
Witt Rings and Brauer Groups Under Multiquadratic Extensions. II DANIEL B
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 78, 58-90 (1982) Witt Rings and Brauer Groups under Multiquadratic Extensions. II DANIEL B. SHAPIRO* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 JEAN-PIERRE TIGNOL Institut de MathPtnatiques Pure et AppliquPe, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium AND ADRIAN R. WADSWORTH* Department of Mathematics, University of California, San Diego, LaJolla, California 92093 Communicated by Richard G. Swan Received May 8, 1981 0. INTRODUCTION Let F be a field of characteristic not 2, and suppose M is a multiquadratic extension of F. That is, M/F is a finite abelian extension of exponent 2, so that M = F(@) for some finite subgroup G E F/F’, where $ = F - (0). In studying Brauer groups and products of quaternion algebras, the second author was led to consider the homology groups N,(M/F) of -a certain complex gM,F associated to the extension M/F. This complex appears in [ 11, (3.1); 301 and in (1.1) below. The purpose of the present work is to investigate the first homology group N,(M/F) and to exhibit its close connections with quadratic form theory. The higher homology groups are examined in [ 11,301. The field F is said to have property Pi(n) if N,(M/F) = 1 for every multi- quadratic extension M/F with [M:F] Q 2”. Properties P,(l) and P,(2) always hold, but there are examples [29] of fields F for which P,(3) fails. -
Unipotent Flows and Isotropic Quadratic Forms in Positive Characteristic
UNIPOTENT FLOWS AND ISOTROPIC QUADRATIC FORMS IN POSITIVE CHARACTERISTIC A. MOHAMMADI Abstract. The analogous statement to Oppenheim conjecture over a local field of positive characteristic is proved. The dynamical argument is most involved in the case of characteristic 3. 1. Introduction Let Q be a real non-degenerate indefinite quadratic form in n variables. Further assume Q is not proportional to a form with rational coefficients . It was conjectured by Oppenheim in [O31] that if n ≥ 5 then for any ε > 0 there is x ∈ Zn − {0} such that |Q(x)| < ε. Later on in 1946 Davenport stated the conjecture for n ≥ 3, see [DH46]. However the conjecture even in the case n ≥ 3 is usually referred to as Oppenheim conjecture. Note that if the conjecture is proved for n0 then it holds for n > n0. Hence it is enough to show this for n = 3. Note also that the conclusion of the theorem is false for n = 2. Using methods of analytic number theory the aforementioned conjecture was verified for n ≥ 21 and also for diagonal forms in five variables, see [DH46, DR59]. The Oppenheim conjecture in its full generality was finally settled affirmatively by G. A. Margulis in [Mar86]. Margulis actually proved a reformulation of this con- jecture, in terms of closure of orbits of certain subgroup of SL3(R) on the space of unimodular lattices. This reformulation (as is well-known) is due to M. S. Raghu- nathan and indeed is a special case of Raghunathan’s conjecture on the closure of orbits of unipotent groups. S.