Algebraic Groups and Number Theory

Algebraic Groups and Number Theory

Author: Vladimir Platonov

Publisher: Academic Press

Published: 1993-12-07

Total Pages: 629

ISBN-13: 0080874592

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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.


Algebraic Groups

Algebraic Groups

Author: J. S. Milne

Publisher: Cambridge University Press

Published: 2017-09-21

Total Pages: 665

ISBN-13: 1107167485

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Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites.


Linear Algebraic Groups

Linear Algebraic Groups

Author: T.A. Springer

Publisher: Springer Science & Business Media

Published: 2010-10-12

Total Pages: 347

ISBN-13: 0817648402

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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.


Adeles and Algebraic Groups

Adeles and Algebraic Groups

Author: A. Weil

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 137

ISBN-13: 1468491563

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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.


An Introduction to Algebraic Geometry and Algebraic Groups

An Introduction to Algebraic Geometry and Algebraic Groups

Author: Meinolf Geck

Publisher: Oxford University Press

Published: 2013-03-14

Total Pages: 321

ISBN-13: 019967616X

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An accessible text introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields.


Representations of Algebraic Groups

Representations of Algebraic Groups

Author: Jens Carsten Jantzen

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 594

ISBN-13: 082184377X

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Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them.


Linear Algebraic Groups

Linear Algebraic Groups

Author: James E. Humphreys

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 259

ISBN-13: 1468494430

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James E. Humphreys is a distinguished Professor of Mathematics at the University of Massachusetts at Amherst. He has previously held posts at the University of Oregon and New York University. His main research interests include group theory and Lie algebras, and this graduate level text is an exceptionally well-written introduction to everything about linear algebraic groups.


Linear Algebraic Groups

Linear Algebraic Groups

Author: Armand Borel

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 301

ISBN-13: 1461209412

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This revised, enlarged edition of Linear Algebraic Groups (1969) starts by presenting foundational material on algebraic groups, Lie algebras, transformation spaces, and quotient spaces. It then turns to solvable groups, general properties of linear algebraic groups, and Chevally’s structure theory of reductive groups over algebraically closed groundfields. It closes with a focus on rationality questions over non-algebraically closed fields.


Representation Theory of Algebraic Groups and Quantum Groups

Representation Theory of Algebraic Groups and Quantum Groups

Author: Toshiaki Shoji

Publisher: American Mathematical Society(RI)

Published: 2004

Total Pages: 514

ISBN-13:

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A collection of research and survey papers written by speakers at the Mathematical Society of Japan's 10th International Conference. This title presents an overview of developments in representation theory of algebraic groups and quantum groups. It includes papers containing results concerning Lusztig's conjecture on cells in affine Weyl groups.


Number Theory

Number Theory

Author: Helmut Koch

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 390

ISBN-13: 9780821820544

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Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of 'higher congruences' as an important element of 'arithmetic geometry'. Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem. There are a detailed exposition of the theory of Hecke $L$-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory. The book attempts as much as possible to give simple proofs. It can be used by a beginner in algebraic number theory who wishes to see some of the true power and depth of the subject. The book is suitable for two one-semester courses, with the first four chapters serving to develop the basic material. Chapters 6 through 9 could be used on their own as a second semester course.