Table of Contents

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.

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.

Reviews

“The book contains a huge amount of interesting and very well-chosen exercises. … This ‘encyclopedic’ character of the text may play an important role both as a guide to some special topics and as a source of information for both students and those whose research in related fields creates a need to familiarize themselves with the knowledge of the case when quaternion algebras are relevant.” (Juliusz Brzeziński, Mathematical Reviews, September, 2022)