TY  - BOOK
AU  - Voight, John
AB  - 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.
DO  - 10.1007/978-3-030-56694-4
ID  - OAPEN ID: ONIX_20210714_9783030566944_8
KW  - Associative Rings and Algebras
KW  - Group Theory and Generalizations
KW  - Number Theory
KW  - Open Access
KW  - Quaternions
KW  - Quaternion algebras
KW  - Quaternion orders
KW  - Quaternion ideals
KW  - Noncommutative algebra
KW  - Quaternions and quadratic forms
KW  - Ternary quadratic forms
KW  - Simple algebras and involutions
KW  - Lattices and integral quadratic forms
KW  - Hurwitz order
KW  - Quaternion algebras over local fields
KW  - Quaternion algebras over global fields
KW  - Adelic framework
KW  - Idelic zeta functions
KW  - Quaternions hyperbolic geometry
KW  - Quaternions arithmetic groups
KW  - Quaternions arithmetic geometry
KW  - Supersingular elliptic curves
KW  - Abelian surfaces with QM
KW  - Algebra
KW  - Groups & group theory
L1  - https://library.oapen.org/bitstream/id/d830e505-b1da-4321-adf4-6bae964d6e66/978-3-030-56694-4.pdf
LA  - English
LK  - https://library.oapen.org/handle/20.500.12657/50018
PB  - Springer Nature
PY  - 2021
SN  - 9783030566944
TI  - Quaternion Algebrasnull
ER  - 
