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Quaternion algebras

Voight, John.

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Quaternion algebras

Record Type:	Electronic resources : Monograph/item
Title/Author:	Quaternion algebras/ by John Voight.
Author:	Voight, John.
Published:	Cham :Springer International Publishing : : 2021.,
Description:	xxiii, 885 p. :ill., digital ;24 cm.
[NT 15003449]:	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.
Contained By:	Springer Nature eBook
Subject:	Quaternions. -
Online resource:	https://doi.org/10.1007/978-3-030-56694-4
ISBN:	9783030566944

Quaternion algebras
Voight, John.

Quaternion algebras[electronic resource] /by John Voight. - Cham :Springer International Publishing :2021. - xxiii, 885 p. :ill., digital ;24 cm. - Graduate texts in mathematics,2880072-5285 ;. - Graduate texts in mathematics ;288..

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.

Open access.

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 are recapped throughout.

ISBN: 9783030566944

Standard No.: 10.1007/978-3-030-56694-4doiSubjects--Topical Terms:

536600
Quaternions.

LC Class. No.: QA196

Dewey Class. No.: 512.5

Quaternion algebras
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100 1 $a Voight, John. $3 3506162
245 1 0 $a Quaternion algebras $h [electronic resource] / $c by John Voight.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2021.
300 $a xxiii, 885 p. : $b ill., digital ; $c 24 cm.
490 1 $a Graduate texts in mathematics, $x 0072-5285 ; $v 288
505 0 $a 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.
506 $a Open access.
520 $a 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 are recapped throughout.
650 0 $a Quaternions. $3 536600
650 0 $a Functions, Quaternion. $3 2058363
650 1 4 $a Associative Rings and Algebras. $3 897405
650 2 4 $a Group Theory and Generalizations. $3 893889
650 2 4 $a Number Theory. $3 891078
710 2 $a SpringerLink (Online service) $3 836513
773 0 $t Springer Nature eBook
830 0 $a Graduate texts in mathematics ; $v 288. $3 3506163
856 4 0 $u https://doi.org/10.1007/978-3-030-56694-4
950 $a Mathematics and Statistics (SpringerNature-11649)