Quadratic Forms And Quaternion Algebras

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Quaternion Algebras

Author : John Voight
Publisher : Springer Nature
ISBN 13 : 3030566943
Total Pages : 877 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Quaternion Algebras by : John Voight

Download or read book Quaternion Algebras written by John Voight and published by Springer Nature. This book was released on 2021-06-28 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.

Quaternion Orders, Quadratic Forms, and Shimura Curves

Author : Montserrat Alsina
Publisher : American Mathematical Soc.
ISBN 13 : 9780821833599
Total Pages : 232 pages
Book Rating : 4.8/5 (335 download)

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Book Synopsis Quaternion Orders, Quadratic Forms, and Shimura Curves by : Montserrat Alsina

Download or read book Quaternion Orders, Quadratic Forms, and Shimura Curves written by Montserrat Alsina and published by American Mathematical Soc.. This book was released on 2004 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shimura curves are a far-reaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplicationpoints. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss'theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully worked-out examples, preparing the way for further research.

Quaternion Orders, Quadratic Forms, and Shimura Curves

Author : Montserrat Alsina and Pilar Bayer
Publisher : American Mathematical Soc.
ISBN 13 : 0821869833
Total Pages : 216 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quaternion Orders, Quadratic Forms, and Shimura Curves by : Montserrat Alsina and Pilar Bayer

Download or read book Quaternion Orders, Quadratic Forms, and Shimura Curves written by Montserrat Alsina and Pilar Bayer and published by American Mathematical Soc.. This book was released on with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shimura curves are a far-reaching generalization of the classical modular curves. They lie at the crossroads of many areas, including complex analysis, hyperbolic geometry, algebraic geometry, algebra, and arithmetic. This monograph presents Shimura curves from a theoretical and algorithmic perspective. The main topics are Shimura curves defined over the rational number field, the construction of their fundamental domains, and the determination of their complex multiplication points. The study of complex multiplication points in Shimura curves leads to the study of families of binary quadratic forms with algebraic coefficients and to their classification by arithmetic Fuchsian groups. In this regard, the authors develop a theory full of new possibilities that parallels Gauss' theory on the classification of binary quadratic forms with integral coefficients by the action of the modular group. This is one of the few available books explaining the theory of Shimura curves at the graduate student level. Each topic covered in the book begins with a theoretical discussion followed by carefully worked-out examples, preparing the way for further research. Titles in this series are co-published with the Centre de Recherches Mathématiques.

Quadratic and Higher Degree Forms

Author : Krishnaswami Alladi
Publisher : Springer Science & Business Media
ISBN 13 : 1461474884
Total Pages : 303 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Quadratic and Higher Degree Forms by : Krishnaswami Alladi

Download or read book Quadratic and Higher Degree Forms written by Krishnaswami Alladi and published by Springer Science & Business Media. This book was released on 2013-08-13 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last decade, the areas of quadratic and higher degree forms have witnessed dramatic advances. This volume is an outgrowth of three seminal conferences on these topics held in 2009, two at the University of Florida and one at the Arizona Winter School. The volume also includes papers from the two focused weeks on quadratic forms and integral lattices at the University of Florida in 2010.Topics discussed include the links between quadratic forms and automorphic forms, representation of integers and forms by quadratic forms, connections between quadratic forms and lattices, and algorithms for quaternion algebras and quadratic forms. The book will be of interest to graduate students and mathematicians wishing to study quadratic and higher degree forms, as well as to established researchers in these areas. Quadratic and Higher Degree Forms contains research and semi-expository papers that stem from the presentations at conferences at the University of Florida as well as survey lectures on quadratic forms based on the instructional workshop for graduate students held at the Arizona Winter School. The survey papers in the volume provide an excellent introduction to various aspects of the theory of quadratic forms starting from the basic concepts and provide a glimpse of some of the exciting questions currently being investigated. The research and expository papers present the latest advances on quadratic and higher degree forms and their connections with various branches of mathematics.

Quadratic Forms, Linear Algebraic Groups, and Cohomology

Author : Skip Garibaldi
Publisher : Springer Science & Business Media
ISBN 13 : 1441962115
Total Pages : 344 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Quadratic Forms, Linear Algebraic Groups, and Cohomology by : Skip Garibaldi

Download or read book Quadratic Forms, Linear Algebraic Groups, and Cohomology written by Skip Garibaldi and published by Springer Science & Business Media. This book was released on 2010-07-16 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.

Compositions of Quadratic Forms

Author : Daniel B. Shapiro
Publisher : Walter de Gruyter
ISBN 13 : 3110824833
Total Pages : 433 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Compositions of Quadratic Forms by : Daniel B. Shapiro

Download or read book Compositions of Quadratic Forms written by Daniel B. Shapiro and published by Walter de Gruyter. This book was released on 2011-06-24 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

The Algebraic Theory of Quadratic Forms

Author : Tsit-Yuen Lam
Publisher : Addison-Wesley
ISBN 13 : 9780805356663
Total Pages : 344 pages
Book Rating : 4.3/5 (566 download)

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Book Synopsis The Algebraic Theory of Quadratic Forms by : Tsit-Yuen Lam

Download or read book The Algebraic Theory of Quadratic Forms written by Tsit-Yuen Lam and published by Addison-Wesley. This book was released on 1980 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Quadratic Forms -- Algebra, Arithmetic, and Geometry

Author : Ricardo Baeza
Publisher : American Mathematical Soc.
ISBN 13 : 0821846485
Total Pages : 424 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Quadratic Forms -- Algebra, Arithmetic, and Geometry by : Ricardo Baeza

Download or read book Quadratic Forms -- Algebra, Arithmetic, and Geometry written by Ricardo Baeza and published by American Mathematical Soc.. This book was released on 2009-08-14 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents a collection of articles that are based on talks delivered at the International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms held in Frutillar, Chile in December 2007. The theory of quadratic forms is closely connected with a broad spectrum of areas in algebra and number theory. The articles in this volume deal mainly with questions from the algebraic, geometric, arithmetic, and analytic theory of quadratic forms, and related questions in algebraic group theory and algebraic geometry.

Bilinear Algebra

Author : Kazimierz Szymiczek
Publisher : Routledge
ISBN 13 : 1351464213
Total Pages : 496 pages
Book Rating : 4.3/5 (514 download)

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Book Synopsis Bilinear Algebra by : Kazimierz Szymiczek

Download or read book Bilinear Algebra written by Kazimierz Szymiczek and published by Routledge. This book was released on 2017-11-22 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Giving an easily accessible elementary introduction to the algebraic theory of quadratic forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading topics include the geometry of bilinear spaces, classification of bilinear spaces up to isometry depending on the ground field, formally real fields, Pfister forms, the Witt ring of an arbitrary field (characteristic two included), prime ideals of the Witt ring, Brauer group of a field, Hasse and Witt invariants of quadratic forms, and equivalence of fields with respect to quadratic forms. Problem sections are included at the end of each chapter. There are two appendices: the first gives a treatment of Hasse and Witt invariants in the language of Steinberg symbols, and the second contains some more advanced problems in 10 groups, including the u-invariant, reduced and stable Witt rings, and Witt equivalence of fields.

Quadratic and Hermitian Forms

Author : McMaster University
Publisher : American Mathematical Soc.
ISBN 13 : 9780821860083
Total Pages : 362 pages
Book Rating : 4.8/5 (6 download)

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Book Synopsis Quadratic and Hermitian Forms by : McMaster University

Download or read book Quadratic and Hermitian Forms written by McMaster University and published by American Mathematical Soc.. This book was released on 1984 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains the proceedings of the 1983 Seminar on Quadratic and Hermitian Forms held at McMaster University, July 1983. Between 1945 and 1965, most of the work in quadratic (and hermitian) forms took place in arithmetic theory (M Eichler, M Kneser, O T O'Meara).

Introduction to Quadratic Forms over Fields

Author : Tsit-Yuen Lam
Publisher : American Mathematical Soc.
ISBN 13 : 0821810952
Total Pages : 577 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Introduction to Quadratic Forms over Fields by : Tsit-Yuen Lam

Download or read book Introduction to Quadratic Forms over Fields written by Tsit-Yuen Lam and published by American Mathematical Soc.. This book was released on 2005 with total page 577 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new version of the author's prizewinning book, Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., 1973), gives a modern and self-contained introduction to the theory of quadratic forms over fields of characteristic different from two. Starting with few prerequisites beyond linear algebra, the author charts an expert course from Witt's classical theory of quadratic forms, quaternion and Clifford algebras, Artin-Schreier theory of formally real fields, and structural theorems on Witt rings, to the theory of Pfister forms, function fields, and field invariants. These main developments are seamlessly interwoven with excursions into Brauer-Wall groups, local and global fields, trace forms, Galois theory, and elementary algebraic K-theory, to create a uniquely original treatment of quadratic form theory over fields. Two new chapters totaling more than 100 pages have been added to the earlier incarnation of this book to take into account some of the newer results and more recent viewpoints in the area. As is characteristic of this author's expository style, the presentation of the main material in this book is interspersed with a copious number of carefully chosen examples to illustrate the general theory. This feature, together with a rich stock of some 280 exercises for the thirteen chapters, greatly enhances the pedagogical value of this book, both as a graduate text and as a reference work for researchers in algebra, number theory, algebraic geometry, algebraic topology, and geometric topology.

Introduction to Quadratic Forms

Author : Onorato Timothy O’Meara
Publisher : Springer
ISBN 13 : 366241922X
Total Pages : 354 pages
Book Rating : 4.6/5 (624 download)

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Book Synopsis Introduction to Quadratic Forms by : Onorato Timothy O’Meara

Download or read book Introduction to Quadratic Forms written by Onorato Timothy O’Meara and published by Springer. This book was released on 2013-12-01 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Introduction to Quadratic Forms

Author : Onorato Timothy O'Meara
Publisher :
ISBN 13 :
Total Pages : 360 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Introduction to Quadratic Forms by : Onorato Timothy O'Meara

Download or read book Introduction to Quadratic Forms written by Onorato Timothy O'Meara and published by . This book was released on 1963 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Quadratic and Hermitian Forms

Author : W. Scharlau
Publisher : Springer Science & Business Media
ISBN 13 : 3642699715
Total Pages : 431 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Quadratic and Hermitian Forms by : W. Scharlau

Download or read book Quadratic and Hermitian Forms written by W. Scharlau and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.

$K$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras

Author : Bill Jacob
Publisher : American Mathematical Soc.
ISBN 13 : 0821803409
Total Pages : 458 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis $K$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras by : Bill Jacob

Download or read book $K$-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras written by Bill Jacob and published by American Mathematical Soc.. This book was released on 1995 with total page 458 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume 2 of two - also available in a set of both volumes.

Recent Advances in Real Algebraic Geometry and Quadratic Forms

Author : Bill Jacob
Publisher : American Mathematical Soc.
ISBN 13 : 0821851543
Total Pages : 416 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Recent Advances in Real Algebraic Geometry and Quadratic Forms by : Bill Jacob

Download or read book Recent Advances in Real Algebraic Geometry and Quadratic Forms written by Bill Jacob and published by American Mathematical Soc.. This book was released on 1994 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers collected here present an up-to-date record of the current research developments in the fields of real algebraic geometry and quadratic forms. Articles range from the technical to the expository and there are also indications to new research directions.

Quadratic and Hermitian Forms over Rings

Author : Max-Albert Knus
Publisher : Springer Science & Business Media
ISBN 13 : 3642754015
Total Pages : 536 pages
Book Rating : 4.6/5 (427 download)

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Book Synopsis Quadratic and Hermitian Forms over Rings by : Max-Albert Knus

Download or read book Quadratic and Hermitian Forms over Rings written by Max-Albert Knus and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new results are obtained, which went unnoticed even for fields, as in the case of involutions on 16-dimensional central simple algebras. The first chapter gives an introduction to the basic definitions and properties of hermitian forms which are used throughout the book.