Quaternion Algebras by John Voight

Book 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 framework 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

About this book :-
"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 Detail :-
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 Author :-
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".

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