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.
KeywordsAssociative Rings and Algebras; Group Theory and Generalizations; Number Theory; Open Access; Quaternions; Quaternion algebras; Quaternion orders; Quaternion ideals; Noncommutative algebra; Quaternions and quadratic forms; Ternary quadratic forms; Simple algebras and involutions; Lattices and integral quadratic forms; Hurwitz order; Quaternion algebras over local fields; Quaternion algebras over global fields; Adelic framework; Idelic zeta functions; Quaternions hyperbolic geometry; Quaternions arithmetic groups; Quaternions arithmetic geometry; Supersingular elliptic curves; Abelian surfaces with QM; Algebra; Groups & group theory
Publication date and place2021
ImprintSpringer International Publishing
SeriesGraduate Texts in Mathematics,
Groups & group theory