Quaternion algebras / John Voight
Voight, John, author.
2021
QA196
Available Online

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Title Quaternion algebras / John Voight
Author Voight, John, author.
ISBN 9783030566944 (electronic bk.)
3030566943 (electronic bk.)
3030566927
9783030566920
DOI https://doi.org/10.1007/978-3-030-56694-4
Published Cham : Springer, 2021.
Language English
Description 1 online resource
Item Number 10.1007/978-3-030-56694-4 doi
Call Number QA196
Dewey Decimal Classification 512/.5
Summary 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Open access.
Digital File Characteristics text file
PDF
Series Graduate texts in mathematics ; 288. 0072-5285
Available in Other Form Quaternion algebras.
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1. Introduction
2. Beginnings
3. Involutions
4. Quadratic Forms
5. Ternary Quadratic Forms
6. Characteristic 2
7. Simple Algebras
8. Simple Algebras and Involutions
9. Lattices and Integral Quadratic Forms
10. Orders
11. The Hurwitz Order
12. Ternary Quadratic Forms Over Local Fields
13. Quaternion Algebras Over Local Fields
14. Quaternion Algebras Over Global Fields
15. Discriminants
16. Quaternion Ideals and Invertability
17. Classes of Quaternion Ideals
18. Picard Group
19. Brandt Groupoids
20. Integral Representation Theory
21. Hereditary and Extremal Orders
22. Ternary Quadratic Forms
23. Quaternion Orders
24. Quaternion Orders: Second Meeting
25. The Eichler Mass Formula
26. Classical Zeta Functions
27. Adelic Framework
28. Strong Approximation
29. Idelic Zeta Functions
30. Optimal Embeddings
31. Selectivity
32. Unit Groups
33. Hyperbolic Plane
34. Discrete Group Actions
35. Classical Modular Group
36. Hyperbolic Space
37. Fundamental Domains
38. Quaternionic Arithmetic Groups
39. Volume Formula
40. Classical Modular Forms
41. Brandt Matrices
42. Supersingular Elliptic Curves
43. Abelian Surfaces with QM.

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