Quaternion Algebras / by John Voight.

Format:

Book

Author/Creator:

Voight, John (Mathematician)

Series:

Graduate Texts in Mathematics, 2197-5612 ; 288

Language:

English

Subjects (All):

Associative rings.

Associative algebras.

Group theory.

Number theory.

Associative Rings and Algebras.

Group Theory and Generalizations.

Number Theory.

Local Subjects:

Associative Rings and Algebras.

Group Theory and Generalizations.

Number Theory.

Physical Description:

1 online resource (877 p.)

Edition:

1st ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Language Note:

English

Summary:

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. 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 arerecapped throughout.

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.

Notes:

Description based upon print version of record.

ISBN:

3-030-56694-3

OCLC:

1258658936