Reduction Theory And Arithmetic Groups
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Reduction Theory And Arithmetic Groups
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Author : Joachim Schwermer
language : en
Publisher: Cambridge University Press
Release Date : 2022-12-15
Reduction Theory And Arithmetic Groups written by Joachim Schwermer and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-15 with Mathematics categories.
Arithmetic groups are generalisations, to the setting of algebraic groups over a global field, of the subgroups of finite index in the general linear group with entries in the ring of integers of an algebraic number field. They are rich, diverse structures and they arise in many areas of study. This text enables you to build a solid, rigorous foundation in the subject. It first develops essential geometric and number theoretical components to the investigations of arithmetic groups, and then examines a number of different themes, including reduction theory, (semi)-stable lattices, arithmetic groups in forms of the special linear group, unipotent groups and tori, and reduction theory for adelic coset spaces. Also included is a thorough treatment of the construction of geometric cycles in arithmetically defined locally symmetric spaces, and some associated cohomological questions. Written by a renowned expert, this book is a valuable reference for researchers and graduate students.
Reduction Theory And Arithmetic Groups
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Author : Joachim Schwermer
language : en
Publisher: Cambridge University Press
Release Date : 2022-12-31
Reduction Theory And Arithmetic Groups written by Joachim Schwermer and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-12-31 with Mathematics categories.
Build a solid foundation in the area of arithmetic groups and explore its inherent geometric and number-theoretical components.
Arithmetic Groups And Reduction Theory
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Author : Armand Borel
language : en
Publisher:
Release Date : 2020
Arithmetic Groups And Reduction Theory written by Armand Borel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Arithmetic groups categories.
Presents papers and lecture notes from four great contributors of the reduction theory: Armond Borel, Roger Godement, Carl Ludwig Siegel and Andre Weil. They reflect their deep knowledge of the subject and their perspectives.
Algebraic Groups And Number Theory Volume 1
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Author : Vladimir Platonov
language : en
Publisher: Cambridge University Press
Release Date : 2023-08-31
Algebraic Groups And Number Theory Volume 1 written by Vladimir Platonov and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-31 with Mathematics categories.
The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.
Finiteness Properties Of Arithmetic Groups Acting On Twin Buildings
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Author : Stefan Witzel
language : en
Publisher: Springer
Release Date : 2014-07-16
Finiteness Properties Of Arithmetic Groups Acting On Twin Buildings written by Stefan Witzel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-16 with Mathematics categories.
Providing an accessible approach to a special case of the Rank Theorem, the present text considers the exact finiteness properties of S-arithmetic subgroups of split reductive groups in positive characteristic when S contains only two places. While the proof of the general Rank Theorem uses an involved reduction theory due to Harder, by imposing the restrictions that the group is split and that S has only two places, one can instead make use of the theory of twin buildings.
Arithmetic Groups And Their Generalizations
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Author : Lizhen Ji
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Arithmetic Groups And Their Generalizations written by Lizhen Ji and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
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.
Introduction To Arithmetic Groups
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Author : Armand Borel
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-11-07
Introduction To Arithmetic Groups written by Armand Borel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-07 with Education categories.
Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.
Lecture Notes Prepared In Connection With The Summer Institute On Algebraic Groups And Discontinuous Subgroups Held At The University Of Colorado Boulder Colorado July 5 August 6 1965
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Author :
language : en
Publisher:
Release Date : 1965
Lecture Notes Prepared In Connection With The Summer Institute On Algebraic Groups And Discontinuous Subgroups Held At The University Of Colorado Boulder Colorado July 5 August 6 1965 written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1965 with Automorphic functions categories.
The principal developments in the arithmetic aspects of algebraic groups are included in the following five major topics: Algebraic linear groups and arithmetic groups, including a survey of the structure and classification theory over arbitrary fields, groups over Z, the maximal compact subgroups of p-adic groups, and reduction theory for arithmetic groups. Adelization, Tamagawa numbers, the Siegel-Weil formula, Galois cohomology, and approximation theorems. Automorphic forms, the spectral decomposition of L-squared (G/gamma), Eisenstein series, interpretation of a Ramanujan conjecture. The compactification and projective imbedding of arithmetically defined quotients of bounded symmetric domains, partial desingularization of the compactification, fiber systems of polarized abelian varieties, fields of moduli, zeta-functions, theta-functions, L-functions, Fourier-Jacobi series, symplectic representations. Vector-valued cohomology and rigidity theorems. (Author).
Introduction To Arithmetic Groups
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Author : Dave Witte Morris
language : en
Publisher:
Release Date : 2015-04-24
Introduction To Arithmetic Groups written by Dave Witte Morris and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-04-24 with Mathematics categories.
This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n, Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background. A PDF file of the book is available on the internet. This inexpensive printed edition is for readers who prefer a hardcopy.
Algebraic Groups And Number Theory
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Author : Vladimir Platonov
language : en
Publisher: Academic Press
Release Date : 1993-12-07
Algebraic Groups And Number Theory written by Vladimir Platonov and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-12-07 with Mathematics categories.
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.