Algebraic Groups, Quadratic Forms and Related Topics
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The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Essential Dimension, Spinor Groups and Quadratic Forms 3
ESSENTIAL DIMENSION, SPINOR GROUPS, AND QUADRATIC FORMS PATRICK BROSNANy, ZINOVY REICHSTEINy, AND ANGELO VISTOLIz Abstract. We prove that the essential dimension of the spinor group Spinn grows exponentially with n and use this result to show that qua- dratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected. 1. Introduction Let K be a field of characteristic different from 2 containing a square root of −1, W(K) be the Witt ring of K and I(K) be the ideal of classes of even-dimensional forms in W(K); cf. [Lam73]. By abuse of notation, we will write q 2 Ia(K) if the Witt class on the non-degenerate quadratic form q defined over K lies in Ia(K). It is well known that every q 2 Ia(K) can be expressed as a sum of the Witt classes of a-fold Pfister forms defined over K; see, e.g., [Lam73, Proposition II.1.2]. If dim(q) = n, it is natural to ask how many Pfister forms are needed. When a = 1 or 2, it is easy to see that n Pfister forms always suffice; see Proposition 4.1. In this paper we will prove the following result, which shows that the situation is quite different when a = 3. Theorem 1.1. Let k be a field of characteristic different from 2 and n ≥ 2 be an even integer. Then there is a field extension K=k and an n-dimensional quadratic form q 2 I3(K) with the following property: for any finite field extension L=K of odd degree qL is not Witt equivalent to the sum of fewer than 2(n+4)=4 − n − 2 7 3-fold Pfister forms over L. -
From Simple Algebras to the Block-Kato Conjecture Alexander Merkurjev
Public lecture From simple algebras to the Block-Kato conjecture Alexander Merkurjev Professor at the University of California, Los Angeles Guest of the Atlantic Algebra Centre of MUN Distinguished Lecturer Atlantic Association for Research in the Mathematical Sciences Abstract. Hamilton’s quaternion algebra over the real numbers was the first nontrivial example of a simple algebra discovered in 1843. I will review the history of the study of simple algebras over arbitrary fields that culminated in the proof of the Bloch-Kato Conjecture by the Fields Medalist Vladimir Voevodsky. Time and Place The lecture will take place on June 15, 2016, on St. John’s campus of Memorial University of Newfound- land, in Room SN-2109 of Science Building. Beginning at 7 pm. Awards and Distinctions of the Speaker In 1986, Alexander Merkurjev was a plenary speaker at the International Congress of Mathematicians in Berkeley, California. His talk was entitled "Milnor K-theory and Galois cohomology". In 1994, he gave an invited plenary talk at the 2nd European Congress of Mathematics in Budapest, Hungary. In 1995, he won the Humboldt Prize, a prestigious international prize awarded to the renowned scholars. In 2012, he won the Cole Prize in Algebra, for his fundamental contributions to the theory of essential dimension. Short overview of scientific achievements The work of Merkurjev focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K- theory, and central simple algebras. In the early 1980s, he proved a fundamental result about the struc- ture of central simple algebras of period 2, which relates the 2-torsion of the Brauer group with Milnor K- theory. -
Essential Dimension of the Spin Groups in Characteristic 2
Essential dimension of the spin groups in characteristic 2 Burt Totaro The essential dimension of an algebraic group G is a measure of the number of parameters needed to describe all G-torsors over all fields. A major achievement of the subject was the calculation of the essential dimension of the spin groups over a field of characteristic not 2, started by Brosnan, Reichstein, and Vistoli, and completed by Chernousov, Merkurjev, Garibaldi, and Guralnick [3, 4, 7], [18, Theorem 9.1]. In this paper, we determine the essential dimension of the spin group Spin(n) for n ≥ 15 over an arbitrary field (Theorem 2.1). We find that the answer is the same in all characteristics. In contrast, for the groups O(n) and SO(n), the essential dimension is smaller in characteristic 2, by Babic and Chernousov [1]. In characteristic not 2, the computation of essential dimension can be phrased to use a natural finite subgroup of Spin(2r + 1), namely an extraspecial 2-group, a central extension of (Z=2)2r by Z=2. A distinctive feature of the argument in characteristic 2 is that the analogous subgroup is a finite group scheme, a central r r extension of (Z=2) × (µ2) by µ2, where µ2 is the group scheme of square roots of unity. In characteristic not 2, Rost and Garibaldi computed the essential dimension of Spin(n) for n ≤ 14 [6, Table 23B], where case-by-case arguments seem to be needed. We show in Theorem 4.1 that for n ≤ 10, the essential dimension of Spin(n) is the same in characteristic 2 as in characteristic not 2. -
Homage to Daniel Gray Quillen
J. K-Theory 8 (2011), 1–1 ©2011 ISOPP doi:10.1017/is011007014jkt163 Homage to Daniel Gray Quillen Daniel Gray Quillen passed away on April 30 of this year, after a long illness. More than anyone else, he was responsible for creating the subject of algebraic K-theory as it is pursued today, and for demonstrating its power and elegance. He also made fundamental contributions to many other aspects of mathematics: rational homotopy, model categories, formal groups, and cyclic homology, to mention a few. All of the ideas he has developed will survive him and give him the stature of a great mathematician of the 20th century. Many mathematicians including all of the members of our Board were greatly inspired and influenced by his vision, his teaching, and his writing. As editors devoted to a subject that Quillen largely created, we are highly appreciative of his crucial support for the journal ‘K-Theory’ and its successor, the ‘Journal of K-Theory’, and of all he has done for our area of mathematics. He will be greatly missed and fondly remembered. The Editorial Board of the Journal of K-Theory: TONY BAK,PAUL BALMER,SPENCER BLOCH,GUNNAR CARLSSON,ALAIN CONNES,WILLIE CORTINAS,ERIC FRIEDLANDER,MAX KAROUBI,GEN- NADI KASPAROV,ALEXANDER MERKURJEV,AMNON NEEMAN,TIM PORTER, JONATHAN ROSENBERG,MARCO SCHLICHTING,ANDREI SUSLIN,GUOPING TANG,VLADIMIR VOEVODSKY,CHUCK WEIBEL,GUOLIANG YU Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.93, on 26 Sep 2021 at 15:22:09, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. -
View This Volume's Front and Back Matter
http://dx.doi.org/10.1090/gsm/090 Basic Quadratic Forms This page intentionally left blank Basic Quadratic Forms Larr y J. Gerstei n Graduate Studies in Mathematics Volume 90 m"zzr&ie. American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz 2000 Mathematics Subject Classification. Primary llExx, 12Exx, 15-XX. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-90 Library of Congress Cataloging-in-Publication Data Gerstein, Larry J. Basic quadratic forms / Larry J. Gerstein. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 90) Includes bibliographical references and index. ISBN 978-0-8218-4465-6 (alk. paper) 1. Forms, Quadratic. 2. Equations, Quadratic. 3. Number theory. I. Title. QA243.G47 2008 512.7'4-dc22 2007062041 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
Journal of K-Theory
Volume 6 Issue 3 ISSN: 1865-2433 Journal of K-Theory K-theory and its Applications in Algebra, Geometry, Analysis & Topology Special Issue in honor of Andrei Suslin on his 60th birthday A. Bak Part II A. Connes P. Balmer M. Karoubi S. J. Bloch G. G. Kasparov G. E. Carlsson Geometry A. S. Merkurjev G y e r t o e m m e t o r e y G K-theory Algebra Analysis K A y a n r - r t a o b h l e y e e s h g o l t i r s - A y K A n s a i A a s l r y l g b e K y - r t h o e T y o g p o o l l o o p g o y T T o y p g o o E. Friedlander l G. Tang A. Neeman V. Voevodsky T. Porter C. Weibel J. Rosenberg G. Yu A. A. Suslin Downloaded from https://www.cambridge.org/core. IP address: 170.106.202.126, on 29 Sep 2021 at 23:39:20, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1865243310000577 Editorial Board Anthony Bak* (University of Bielefeld) Tim Porter (Bangor University) [email protected] [email protected] Paul Balmer (UCLA) Jonathan M. Rosenberg* (University of Maryland) [email protected] [email protected] Spencer Bloch (University of Chicago) Andrei Suslin (Northwestern University) [email protected] [email protected] Gunnar Carlsson (Stanford University) Guoping Tang (Graduate University of Chinese Academy [email protected] of Sciences) [email protected] Alain Connes (IHES) [email protected] Vladimir Voevodsky (Institute of Advanced Study) [email protected] Eric Friedlander (Northwestern University) [email protected] Charles Weibel* -
Pfister Forms and Their Applications the Purpose of This Note Is to Present Results Which Follow from the Theory of Pfister Form
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF NUMBER THEORY 5, 367-378 (1973) Pfister Forms and Their Applications RICHARD ELMAN AND T. Y. LAM* Department of Mathematics, University of California, Berkeley, Carifornia 94720 PRESENTED AT THE QUADRATIC FORMS CONFERENCE, BATON ROUGE, LOUISIANA, MARCH 27-30, 1972, AND DEDICATED TO THE MEMORY OF LOUIS JOEL MORDELL This note investigates the properties of the 2”-dimensional quadratic forms @L, <I, a,>, called n-fold Ptister forms. Utilizing these properties, various applications are made to k-theory of fields, and field invariants. In particular, the set of orderings of a field and the maximum dimension of anisotropic forms with everywhere zero signature are investigated. Details will appear elsewhere. 1. INTRODUCTION AND TERMINOLOGY The purpose of this note is to present results which follow from the theory of Pfister forms. In particular, applications to various invariants are given. Sections 2 and 3 set up the basic machinery of Plister forms. Of central importance is the concept of “linkage,” investigation of which occupies the major part of Section 3. In Section 4, we give a few immediate applications toward answering a question of Milnor. Section 5 is concerned with studying the structure of formally real fields. We investigate separation properties of the set of orderings of such fields, and relate these separation properties to linkage of Pfister forms. Section 6 introduces the “general” u-invariant, u(F), a slightly modified version of the u-invariant in the literature. Extending a result of Kneser, we show that for any field F, u(F) < / F/F” I. -
Professor Alexander Merkurjev Report
Atlantic Association for Research in the Mathematical Sciences Atlantic Algebra Centre (AAC) Network of Ontario Lie Theorists (NOLT) AARMS Distinguished Lecturer Professor Alexander Merkurjev June 13 –17, 2016 Memorial University of Newfoundland St John’s, Newfoundland and Labrador Report Alexander Merkurjev (University of California – Los Angeles) is an illustrious mathematician working in the area of Algebra. The list of his achievements is long and includes the following. In 1986, Alexander Merkurjev was a plenary speaker at the International Congress of Mathematicians in Berkeley, California. His talk was entitled "Milnor K-theory and Galois cohomology". In 1994, he gave an invited plenary talk at the 2nd European Congress of Mathematics in Budapest, Hungary. In 1995, he won the Humboldt Prize, a prestigious international prize awarded to renowned scholars. In 2012, he won the Cole Prize in Algebra, for his fundamental contributions to the theory of essential dimension. In 2016, Alexander Merkurjev was named an AARMS Distinguished Lecturer. His visit to Atlantic Canada was in the period June 12 – 18, 2016. During this visit, he delivered two lectures on St. John’s campus of Memorial University of Newfoundland (MUN). The first one, on June 14, was a plenary lecture for the participants of the International Workshop “Hopf Algebras, Algebraic Groups and Related Structures” organized by Atlantic Algebra Centre (AAC) and the Network of Ontario Lie Theorists (NOLT). The title was “Rationality problem for classifying spaces of algebraic groups”. The lecture was also attended by some students and faculty of the Department of Mathematics and Statistics of MUN, to the total of 40 people. -
Mathematisches Forschungsinstitut Oberwolfach Quadratic Forms and Linear Algebraic Groups
Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2009 DOI: 10.4171/OWR/2009/25 Quadratic Forms and Linear Algebraic Groups Organised by Detlev Hoffmann (Nottingham) Alexander S. Merkurjev (Los Angeles) Jean-Pierre Tignol (Louvain-la-Neuve) May 10th – May 16th, 2009 Abstract. Topics discussed at the workshop Quadratic forms and linear algebraic groups included besides the algebraic theory of quadratic and Her- mitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic as- pects pertaining to the theory of quadratic forms over function fields over local fields. Mathematics Subject Classification (2000): 11Exx, 12Gxx, 14C15, 14C25, 14C35, 14E08, 14G20, 14J50, 14M17, 16K20, 16K50, 17B25, 18E30, 19Gxx, 20Gxx, 22E47. Introduction by the Organisers The workshop was organized by Detlev Hoffmann (Nottingham), Alexander Merkurjev (Los Angeles), and Jean-Pierre Tignol (Louvain-la-Neuve), and was at- tended by 50 participants. Funding from the Leibniz Association within the grant “Oberwolfach Leibniz Graduate Students” (OWLG) provided support toward the participation of five young researchers. Additionally, the “US Junior Oberwolfach Fellows” program of the US National Science Foundation funded travel expenses for two post docs from the USA. The workshop followed a long tradition of Oberwolfach meetings on algebraic theory of quadratic forms and related structures. In the last 15 years the algebraic theory of quadratic forms was greatly influenced by methods of algebraic geometry and linear algebraic groups. The schedule of the meeting comprised 21 lectures of 45 minutes each. High- lights of the conference include recent progress of the application of the patching 1376 Oberwolfach Report 25 principle, progress on the Grothendieck-Serre conjecture, and the solution of the problem on excellent connections in the motive of a quadratic form. -
Algebraic Tori and Essential Dimension
University of California Los Angeles Algebraic Tori and Essential Dimension A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Anthony Michael Ruozzi 2012 c Copyright by Anthony Michael Ruozzi 2012 Abstract of the Dissertation Algebraic Tori and Essential Dimension by Anthony Michael Ruozzi Doctor of Philosophy in Mathematics University of California, Los Angeles, 2012 Professor Alexander Merkurjev, Chair Interest in essential dimension problems has been growing over the past decade. In part, it is because the idea of essential dimension captures quite elegantly the problem of parametrizing a wide range of algebraic objects. But perhaps more, it is because the study of essential dimension requires most of the algebraic arsenal. What began as a problem in Galois co- homology and representation theory now has connections to versal torsors, stacks, motives, birational geometry, and invariant theory. This exposition will focus on just a small bit of this theory: algebraic tori and how they can be used to help us calculate the essential p-dimension for PGLn. ii The dissertation of Anthony Michael Ruozzi is approved. Amit Sahai Paul Balmer Richard Elman Alexander Merkurjev, Committee Chair University of California, Los Angeles 2012 iii Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Basic Notations . 2 2 Groups and Algebras :::::::::::::::::::::::::::::::: 3 2.1 p-closed fields . 3 2.2 Central Simple Algebras . 4 2.2.1 Etale´ Algebras . 5 2.2.2 Division Algebras of Degree ps ...................... 5 2.3 Algebraic Groups . 8 2.3.1 Groups of Multiplicative Type . 9 2.3.2 Galois Cohomology . -
Report on the BIRS Workshop Quadratic Forms, Algebraic Groups, and Galois Cohomology October 4 to 9, 2003
Report on the BIRS workshop Quadratic Forms, Algebraic Groups, and Galois Cohomology October 4 to 9, 2003 Richard Elman, Alexander Merkurjev, Jan Min´aˇc, Carl Riehm The algebraic theory of quadratic forms began with the seminal paper of E. Witt [31] in 1937, where what are now called \Witt's Theorem" and the \Witt ring" first appeared. But it was not until a remarkable series of papers [19] by A.Pfister in the mid-1960s that the theory was transformed into a significant field in its own right. The period 1965 to 1980 can be considered the “first phase" of the algebraic theory of quadratic forms and is well documented in the books of T.-Y.Lam [13] (1973) and W.Scharlau [25] (1985). Lam's book itself came at a critical time and greatly influenced the development and popularity of the field. In 1981 another phase began with the first use of sophisticated techniques from outside the field—in this case algebraic geometry|by A.S.Merkurjev, who proved a long-standing conjecture of A.A.Albert on a presentation for the exponent two subgroup of the Brauer group. This answered the first open case of the Milnor conjecture of 1970 [17] on the relationship between algebraic K-theory, the Brauer group and the Witt ring. As the preeminent open problem for many years in the algebraic theory of quadratic forms, the Milnor conjecture exerted a profound influence on the subject. This work was extended shortly thereafter (1982) in a paper by Merkurjev and A.A.Suslin [15], showing that when the field F contains the nth roots of unity, the n- torsion in the Brauer group of F is isomorphic to the algebraic K-group K2(F )=nK2(F ).