Quaternion Algebras by John Voight

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About this book :-
This graduate-level textbook gives a clear, step-by-step introduction to quaternion algebras (a class of non-commutative algebraic structures). It developed algebra, arithmetic, analysis, geometry, and arithmetic geometry of quaternion algebras. It's well-structured, includes helpful introductions to advanced topics, and serves both as a learning tool and a long-term reference. This text is devided in five parts Algebra, Arithmetic, Analysis, Arithmetic Geometry and Geometry & Topology. The text 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.

Book Detail :-
This book has following details information.
Title: Quaternion Algebras by John Voight by NA
Publisher: Springer
Series: freeCompBooks
Year: 2022
Pages: 908
Type: PDF
Language: English
ISBN-10 #: 3030574679
ISBN-13 #: 978-3030574673
Country: Pakistan
License: CC BY-NC 4.0
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About Author :-
The author NA NA

Book Contents :-
conver the following topics.
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

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