Quaternion Algebras by John Voight
Quaternion Algebras - Table of Contents
1. Introduction
Part-I Algebra
2. Beginnings
3. Involutions
4. Quadratic Forms
5. Ternary Quadratic Forms and Quaternion Algebras
6. Characteristic
7. Simple Algebras
8. Simple Algebras and Involutions
Part-II Arithmetic
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 Invertibility
17. Classes of Quaternion Ideals
18. Two-Sided Ideals and the Picard Group
19. Brandt Groupoids
20. Integral Representation Theory
21. Hereditary and Extremal Orders
22. Quaternion Orders and Ternary Quadratic Forms
23. Quaternion Orders
24. Quaternion Orders: Second Meeting
Part-III Analysis
25. The Eichler Mass Formula
26. Classical Zeta Functions
27. Adelic Frame work
28. Strong Approximation
29. Idelic Zeta Functions
30. Optimal Embeddings
31. Selectivity
Part-IV Geometry and Topology
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
Part-V Arithmetic Geometry
40. Classical Modular Forms
41. Brandt Matrices
42. Supersingular Elliptic Curves
43. QM Abelian Surfaces
What You Will Learn in Quaternion Algebras
"Quaternion Algebras" by John Voight is a clear and modern introduction to the theory of quaternion algebras, written for graduate students and researchers. The book builds a strong foundation in "abstract algebra", carefully explaining how quaternion algebras fit into the broader theory of central simple algebras. Starting from classical Hamilton quaternions, Voight gradually moves toward deeper structural results over general fields.
The text explores key concepts such as "number theory", local and global fields, ramification, norms, and traces. It connects algebraic theory with arithmetic applications, helping readers understand how quaternion algebras interact with modular forms and other advanced topics. Each chapter is written with detailed proofs and logical development, making complex ideas easier to follow.
One of the book’s strengths is its balance between theory and practice. Through examples and exercises, readers gain insight into "field theory", "algebraic structures", and "arithmetic geometry". Overall, it serves as both a learning guide and a reliable reference for advanced mathematical study.
Book Details & Specifications
Title:
Quaternion Algebras by John Voight
Publisher:
Springer
Year:
2022
Pages:
908
Type:
PDF
Language:
English
ISBN-10 #:
3030574679
ISBN-13 #:
978-3030574673
License:
CC BY-NC 4.0
Amazon:
Amazon
About the Author: John Voight
The author John Voight
is a respected scholar in "number theory", "algebra", and arithmetic research. As a professor at Dartmouth College, he focuses on deep mathematical structures, especially in "quaternion algebras" and modular forms. His academic work blends theory with practical computation, helping modern mathematicians understand complex algebraic systems. He is best known for his book Quaternion Algebras, a clear and structured guide to advanced "arithmetic geometry" and "computational mathematics".
Read or Downloadable Quaternion Algebras
Free Abstract Algebra Books & Textbooks PDF