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.
More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout..
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.
Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment.
Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored.
Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields.
Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces.
Applications of unit groups of Quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants.
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.
An in-depth exploration of the arithmetic of Quaternion Algebras and orders follows.
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.
Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike.
Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature.
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
