Quaternion Algebras

This open-access textbook provides a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, with applications in various areas of mathematics. It is written for graduate students and covers basic introductions, arithmetic aspects, analytic approaches, applications to hyperbolic geometry and low-dimensional topology, and arithmetic geometry. The book assumes familiarity with algebraic number theory, commutative algebra, linear algebra, topology, and complex analysis, and provides pathways for exploration in different directions.

Format: Paperback / softback
Length: 885 pages
Publication date: 29 June 2022
Publisher: Springer Nature Switzerland AG

This open-access textbook offers a comprehensive exploration of the arithmetic theory of quaternion algebras and orders, a subject with applications across various branches of mathematics. Written with accessibility in mind for graduate students, this text consolidates and synthesizes results from across the literature, providing a valuable resource for both students and researchers. Divided into five parts, the book begins with a foundational introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. The subsequent sections delve into the intricate arithmetic of quaternion algebras and orders, offering a deep understanding of their properties and applications. The third part explores analytic aspects, beginning with zeta functions and progressing to an idelic approach, which provides a pathway from local to global, encompassing strong approximation techniques. The fourth part showcases the applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology, elucidating the connections between geometric and topological properties and arithmetic invariants. Arithmetic geometry concludes the volume, covering quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces.

Quaternion Algebras is a multidisciplinary field that intersects algebra, geometry, and number theory, offering a rich tapestry of knowledge for those interested in these areas. Graduate students embarking on their academic journey will find this textbook an invaluable resource, as it provides a comprehensive foundation upon which to build their understanding of quaternion algebras and orders. Instructors will also find this text highly useful in designing introductory and advanced courses, as it offers a well-structured framework for teaching these complex subjects. Researchers will appreciate the all-encompassing treatment of the topic, which encompasses a wide range of results and perspectives.

Before delving into the details, it is important to have a basic understanding of algebraic number theory and commutative algebra, as these foundational subjects will be referred to throughout the text. Additionally, readers should be familiar with the concept of quaternions, which are four-dimensional vectors that satisfy certain mathematical properties.

The book is organized into five parts, each dedicated to a specific aspect of quaternion algebras and orders. The first part provides a foundational introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields. This section covers topics such as commutative rings, ideals, and modules, which are essential building blocks for quaternion algebras. It also introduces the concept of quaternions, their representation theory, and their basic properties.

The second part delves into the arithmetic of quaternion algebras and orders. This section explores the fundamental properties of quaternion algebras, such as their rings, ideals, and modules, as well as their representation theory. It also discusses the arithmetic operations on quaternion algebras, including addition, multiplication, and division. The third part explores analytic aspects of quaternion algebras and orders. This section introduces zeta functions, which are important tools in the study of quaternion algebras. It also discusses the idelic approach, which provides a global perspective on quaternion algebras by considering their analytic properties. The fourth part showcases the applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology. This section explores the connections between geometric and topological properties and arithmetic invariants, and how quaternion orders can be used to study these topics. The fifth part concludes the book by covering arithmetic geometry, which is a branch of mathematics that studies the arithmetic properties of geometric objects. This section discusses quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces.

Throughout the book, numerous pathways offer explorations in many different directions. These pathways provide a roadmap for readers to delve deeper into specific topics or branches of the field, allowing them to tailor their learning experience to their interests and needs. The unified treatment of the topic makes this book an essential reference for students and researchers alike.

In conclusion, Quaternion Algebras is a comprehensive and accessible textbook that offers a thorough exploration of the arithmetic theory of quaternion algebras and orders. Written with graduate students in mind, this text provides a valuable resource for both students and researchers, offering a well-structured framework for understanding these complex subjects. With numerous pathways and a unified treatment, this book is an essential reference for anyone interested in algebra, geometry, and number theory.

Weight: 1371g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030574673
Edition number: 1st ed. 2021