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Algebraic Groups and Their Birational Invariants

Algebraic Groups and Their Birational Invariants PDF Author: V. E. Voskresenskii
Publisher: American Mathematical Soc.
ISBN: 0821872885
Category : Mathematics
Languages : en
Pages : 234

Book Description
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.

Algebraic Groups and Their Birational Invariants

Algebraic Groups and Their Birational Invariants PDF Author: V. E. Voskresenskii
Publisher: American Mathematical Soc.
ISBN: 0821872885
Category : Mathematics
Languages : en
Pages : 234

Book Description
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.

Algebraic Groups

Algebraic Groups PDF Author: J. S. Milne
Publisher: Cambridge University Press
ISBN: 1316739155
Category : Mathematics
Languages : en
Pages : 665

Book Description
Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

Multiplicative Invariant Theory

Multiplicative Invariant Theory PDF Author: Martin Lorenz
Publisher: Springer Science & Business Media
ISBN: 9783540243236
Category : Mathematics
Languages : en
Pages : 200

Book Description
Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far.. Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori. Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2. The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

Algebraic Groups and Number Theory

Algebraic Groups and Number Theory PDF Author: Vladimir Platonov
Publisher: Cambridge University Press
ISBN: 052111361X
Category : Mathematics
Languages : en
Pages : 379

Book Description
The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.

Arithmetic Groups and Their Generalizations

Arithmetic Groups and Their Generalizations PDF Author: Lizhen Ji
Publisher: American Mathematical Soc.
ISBN: 0821848666
Category : Mathematics
Languages : en
Pages : 282

Book Description
In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n, \mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics. Titles in this series are co-published with International Press, Cambridge, MA.Table of Contents: Introduction; General comments on references; Examples of basic arithmetic groups; General arithmetic subgroups and locally symmetric spaces; Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups; Different completions of $\mathbb{Q}$ and $S$-arithmetic groups over number fields; Global fields and $S$-arithmetic groups over function fields; Finiteness properties of arithmetic and $S$-arithmetic groups; Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients; Compactifications of locally symmetric spaces; Rigidity of locally symmetric spaces; Automorphic forms and automorphic representations for general arithmetic groups; Cohomology of arithmetic groups; $K$-groups of rings of integers and $K$-groups of group rings; Locally homogeneous manifolds and period domains; Non-cofinite discrete groups, geometrically finite groups; Large scale geometry of discrete groups; Tree lattices; Hyperbolic groups; Mapping class groups and outer automorphism groups of free groups; Outer automorphism group of free groups and the outer spaces; References; Index. Review from Mathematical Reviews: ...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.(AMSIP/43.

Harmonic Analysis, Group Representations, Automorphic Forms, and Invariant Theory

Harmonic Analysis, Group Representations, Automorphic Forms, and Invariant Theory PDF Author: Jian-Shu Li
Publisher: World Scientific
ISBN: 981277078X
Category : Mathematics
Languages : en
Pages : 446

Book Description
This volume carries the same title as that of an international conference held at the National University of Singapore, 9-11 January 2006 on the occasion of Roger E. Howe's 60th birthday. Authored by leading members of the Lie theory community, these contributions, expanded from invited lectures given at the conference, are a fitting tribute to the originality, depth and influence of Howe's mathematical work. The range and diversity of the topics will appeal to a broad audience of research mathematicians and graduate students interested in symmetry and its profound applications.

Rationality Problem for Algebraic Tori

Rationality Problem for Algebraic Tori PDF Author: Akinari Hoshi
Publisher: American Mathematical Soc.
ISBN: 1470424096
Category : Mathematics
Languages : en
Pages : 228

Book Description
The authors give the complete stably rational classification of algebraic tori of dimensions and over a field . In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank and is given. The authors show that there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension , and there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension . The authors make a procedure to compute a flabby resolution of a -lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a -lattice is invertible (resp. zero) or not. Using the algorithms, the suthors determine all the flabby and coflabby -lattices of rank up to and verify that they are stably permutation. The authors also show that the Krull-Schmidt theorem for -lattices holds when the rank , and fails when the rank is ...

Buildings, Finite Geometries and Groups

Buildings, Finite Geometries and Groups PDF Author: N.S. Narasimha Sastry
Publisher: Springer Science & Business Media
ISBN: 1461407095
Category : Mathematics
Languages : en
Pages : 348

Book Description
This is the Proceedings of the ICM 2010 Satellite Conference on “Buildings, Finite Geometries and Groups” organized at the Indian Statistical Institute, Bangalore, during August 29 – 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups. The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

Algebraic Groups

Algebraic Groups PDF Author: Yuri Tschinkel
Publisher: Universitätsverlag Göttingen
ISBN: 3938616776
Category : Algebraic varieties
Languages : en
Pages : 168

Book Description


Unramified Brauer Group and Its Applications

Unramified Brauer Group and Its Applications PDF Author: Sergey Gorchinskiy
Publisher: American Mathematical Soc.
ISBN: 1470440725
Category : Mathematics
Languages : en
Pages : 201

Book Description
This book is devoted to arithmetic geometry with special attention given to the unramified Brauer group of algebraic varieties and its most striking applications in birational and Diophantine geometry. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, Weil restriction of scalars, algebraic tori, the Hasse principle, Brauer-Manin obstruction, and étale cohomology. The book contains a detailed presentation of an example of a stably rational but not rational variety, which is presented as series of exercises with detailed hints. This approach is aimed to help the reader understand crucial ideas without being lost in technical details. The reader will end up with a good working knowledge of the Brauer group and its important geometric applications, including the construction of unirational but not stably rational algebraic varieties, a subject which has become fashionable again in connection with the recent breakthroughs by a number of mathematicians.