UID:
kobvindex_HPB1258658936
Format:
1 online resource
ISBN:
9783030566944
,
3030566943
Series Statement:
Graduate texts in mathematics, 288
Content:
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.
Note:
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.
Additional Edition:
Print version: Voight, John. Quaternion algebras. Cham : Springer, 2021 3030566927
Additional Edition:
9783030566920
Language:
English
Keywords:
Electronic books.
;
Electronic books.
DOI:
10.1007/978-3-030-56694-4
URL:
Click here to view book
