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.
