Overview
- Presents a comprehensive, open access treatment of the theory of quaternions, with the eBook free for all readers
- Engages the student reader with an accessible, approachable writing style
- Offers numerous options for constructing introductory and advanced courses
- Encompasses a vast wealth of knowledge to form an essential reference
Part of the book series: Graduate Texts in Mathematics (GTM, volume 288)
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Table of contents (43 chapters)
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Front Matter
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Arithmetic
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Front Matter
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Keywords
- Open Access
- Quaternions
- Quaternion algebras
- Quaternion orders
- Quaternion ideals
- Noncommutative algebra
- Quaternions and quadratic forms
- Ternary quadratic forms
- Simple algebras and involutions
- Lattices and integral quadratic forms
- Hurwitz order
- Quaternion algebras over local fields
- Quaternion algebras over global fields
- Adelic framework
- Idelic zeta functions
- Quaternions hyperbolic geometry
- Quaternions arithmetic groups
- Quaternions arithmetic geometry
- Supersingular elliptic curves
- Abelian surfaces with QM
About this book
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 unitgroups 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 andmotivation are recapped throughout.
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Authors and Affiliations
About the author
John Voight is Professor of Mathematics at Dartmouth College in Hanover, New Hampshire. His research interests lie in arithmetic algebraic geometry and number theory, with a particular interest in computational aspects. He has taught graduate courses in algebra, number theory, cryptography, as well as the topic of this book, quaternion algebras.
Bibliographic Information
Book Title: Quaternion Algebras
Authors: John Voight
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-56694-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s) 2021
Hardcover ISBN: 978-3-030-56692-0Published: 29 June 2021
Softcover ISBN: 978-3-030-57467-3Published: 30 June 2022
eBook ISBN: 978-3-030-56694-4Published: 28 June 2021
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XXIII, 885
Number of Illustrations: 67 b/w illustrations, 2 illustrations in colour
Topics: Associative Rings and Algebras, Group Theory and Generalizations, Number Theory