Author: Vladimir Platonov,Andrei Rapinchuk,Rachel Rowen

Publisher: Academic Press

ISBN: 9780080874593

Category: Mathematics

Page: 614

View: 2430

This milestone work on the arithmetic theory of linear algebraic groups is now available in English for the first time. Algebraic Groups and Number Theory provides the first systematic exposition in mathematical literature of the junction of group theory, algebraic geometry, and number theory. The exposition of the topic is built on a synthesis of methods from algebraic geometry, number theory, analysis, and topology, and the result is a systematic overview ofalmost all of the major results of the arithmetic theory of algebraic groups obtained to date.
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Author: Skip Garibaldi,R. Sujatha,Venapally Suresh

Publisher: Springer Science & Business Media

ISBN: 9781441962119

Category: Mathematics

Page: 348

View: 2753

The invited papers collected in this volume address topics related to the research of Raman Parimala (plenary speaker at the upcoming ICM 2010). These themes focus primarily on the interplay between algebra, number theory, and algebraic geometry. The included contributions cover exciting research in areas such as field patching and a proof of the Serre's Conjecture II for function fields of complex surfaces.
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A Volume of Papers in Honour of the Late R. W. Richardson

Author: T. A. Springer,Roger Wolcott Richardson,G. I. Lehrer,Alan L. Carey,Michael Murray

Publisher: Cambridge University Press

ISBN: 9780521585323

Category: Mathematics

Page: 384

View: 9985

This volume is a unique and comprehensive collection of works by some of the world's leading researchers. Papers on algebraic geometry, algebraic groups, and Lie groups are woven together to form a connection between the study of symmetry and certain algebraic structures. This connection reflects the interests of R. W. Richardson who studied the links between representation theory and the structure and geometry of algebraic groups. In particular, the papers address Kazhdan-Lusztig theory, quantum groups, spherical varieties, symmetric varieties, cohomology of varieties, purity, Schubert geometry, invariant theory and symmetry breaking. For those working on algebraic and Lie groups, this book will be a wealth of fascinating material.
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Author: Akihiko Gyoja,Hiraku Nakajima,Ken-ichi Shinoda,Toshiaki Shoji,Toshiyuki Tanisaki

Publisher: Springer Science & Business Media

ISBN: 9780817646974

Category: Mathematics

Page: 348

View: 7829

Invited articles by top notch experts Focus is on topics in representation theory of algebraic groups and quantum groups Of interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
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Author: V. E. Voskresenskii,V. E. VoskresenskiuI and Boris Kunyavski

Publisher: American Mathematical Soc.

ISBN: 0821872885

Category:

Page: 218

View: 1851

Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a significant revision of the author's book, Algebraic Tori (Nauka, Moscow, 1977)--studies birational properties of linear algebraic groups focusing on arithmetic applications. The main topics are forms and Galois cohomology, the Picard group and the Brauer group, birational geometry of algebraic tori, arithmetic of algebraic groups, Tamagawa numbers, $R$-equivalence, projective toric varieties, invariants of finite transformation groups, and index-formulas. Results and applications are recent. There is an extensive bibliography with additional comments that can serve as a guide for further reading.
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Author: A. Weil

Publisher: Springer Science & Business Media

ISBN: 1468491563

Category: Mathematics

Page: 126

View: 9438

This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel’s work on quadratic forms. Serving as an introduction to the subject, these notes may also provide stimulation for further research.
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Author: Paul B. Garrett

Publisher: CRC Press

ISBN: 9780412063312

Category: Mathematics

Page: 416

View: 8788

Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the "spherical building at infinity" attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment.
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Author: G. P. Hochschild

Publisher: Springer Science & Business Media

ISBN: 1461381142

Category: Mathematics

Page: 267

View: 5100

The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras. It is thus an ideally suitable framework for exhibiting basic algebra in action. To do that is the principal concern of this text. Accordingly, its emphasis is on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of the simple groups and their irreducible representations. In the same spirit, this exposition has been made entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years of graduate study in algebra is pre supposed. The chapter headings should be sufficient indication of the content and organisation of this book. Each chapter begins with a brief announcement of its results and ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references. The references are intended to be merely suggestions for supplementary reading or indications of original sources, especially in cases where these might not be the expected ones. Algebraic group theory has reached a state of maturity and perfection where it may no longer be necessary to re-iterate an account of its genesis. Of the material to be presented here, including much of the basic support, the major portion is due to Claude Chevalley.
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Author: Teresa Crespo,Zbigniew Hajto

Publisher: American Mathematical Soc.

ISBN: 082185318X

Category: Mathematics

Page: 225

View: 6367

Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields.
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Author: Armand Borel

Publisher: American Mathematical Soc.

ISBN: 0821802887

Category: Mathematics

Page: 184

View: 858

This book looks at the development of lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II through IV are devoted to the former, Chapters V through VIII, to the latter.
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Author: Jens Carsten Jantzen

Publisher: American Mathematical Soc.

ISBN: 082184377X

Category: MATHEMATICS

Page: 576

View: 8477

The present book, which is a revised edition of the author's book published in 1987 by Academic Press, is intended to give the reader an introduction to the theory of algebraic representations of reductive algebraic groups. To develop appropriate techniques, the first part of the book is an introduction to the general theory of representations of algebraic group schemes. Here the author describes, among others, such important basic notions as induction functor, cohomology, quotients, Frobenius kernels, and reduction mod $p$. The second part of the book is devoted to the representation theory of reductive algebraic groups. It includes such topics as the description of simple modules, vanishing theorems, the Borel-Bott-Weil theorem and Weyl's character formula, Schubert schemes and line bundles on them. For this revised edition the author added several chapters describing some later developments, among them Schur algebras, Lusztig's conjecture, and Kazhdan-Lusztig polynomials, tilting modules, and representations of quantum groups.
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Transcendence Properties of the Exponential Function in Several Variables

Author: Michel Waldschmidt

Publisher: Springer Science & Business Media

ISBN: 3662115697

Category: Mathematics

Page: 633

View: 8937

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.
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Author: A. Fröhlich,M. J. Taylor,Martin J. Taylor

Publisher: Cambridge University Press

ISBN: 9780521438346

Category: Mathematics

Page: 355

View: 2395

This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.
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Author: T.A. Springer

Publisher: Springer Science & Business Media

ISBN: 0817648402

Category: Mathematics

Page: 334

View: 587

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.
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Diophantine, Computational and Algebraic Aspects. Proceedings of the International Conference held in Eger, Hungary, July 29-August 2, 1996

Author: Kalman Gyoery,Attila Pethoe,Vera T. Sos

Publisher: Walter de Gruyter

ISBN: 3110809796

Category: Mathematics

Page: 612

View: 9808

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.
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Author: Mahir Bilen Can

Publisher: American Mathematical Soc.

ISBN: 1470426013

Category: Algebraic geometry -- Algebraic groups -- Group schemes

Page: 294

View: 8895

This volume contains the proceedings of the 2015 Clifford Lectures on Algebraic Groups: Structures and Actions, held from March 2–5, 2015, at Tulane University, New Orleans, Louisiana. This volume consists of six articles on algebraic groups, including an enhanced exposition of the classical results of Chevalley and Rosenlicht on the structure of algebraic groups; an enhanced survey of the recently developed theory of pseudo-reductive groups; and an exposition of the recently developed operational -theory for singular varieties. In addition, there are three research articles containing previously unpublished foundational results on birational automorphism groups of algebraic varieties; solution of Hermite-Joubert problem over -closed fields; and cohomological invariants and applications to classifying spaces. The old and new results presented in these articles will hopefully become cornerstones for the future development of the theory of algebraic groups and applications. Graduate students and researchers working in the fields of algebraic geometry, number theory, and representation theory will benefit from this unique and broad compilation of fundamental results on algebraic group theory.
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Author: Rolf Berndt,Ralf Schmidt

Publisher: Springer Science & Business Media

ISBN: 3034802838

Category: Mathematics

Page: 213

View: 2707

Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.
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From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory

Author: Benjamin Fine,Anthony M. Gaglione,Gerhard Rosenberger

Publisher: JHU Press

ISBN: 1421411768

Category: Mathematics

Page: 584

View: 8236

Presents a systematic approach to one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, this title begins with familiar topics such as rings, numbers, and groups before introducing more difficult concepts.
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Author: H. Koch

Publisher: Springer Science & Business Media

ISBN: 3642580955

Category: Mathematics

Page: 269

View: 5321

From the reviews: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994 "... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995
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