Quaternion Algebras

1. Front Matter

   Pages i-xxiii

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2. Introduction

   • John Voight

   Pages 1-17Open Access

3. Algebra

   1. Front Matter

      Pages 19-19

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   2. Beginnings

      • John Voight

      Pages 21-34Open Access

   3. Involutions

      • John Voight

      Pages 35-45Open Access

   4. Quadratic forms

      • John Voight

      Pages 47-63Open Access

   5. Ternary quadratic forms and quaternion algebras

      • John Voight

      Pages 65-84Open Access

   6. Characteristic 2

      • John Voight

      Pages 85-94Open Access

   7. Simple algebras

      • John Voight

      Pages 95-121Open Access

   8. Simple algebras and involutions

      • John Voight

      Pages 123-135Open Access

4. Arithmetic

   1. Front Matter

      Pages 137-137

      PDF

   2. Lattices and integral quadratic forms

      • John Voight

      Pages 139-154Open Access

   3. Orders

      • John Voight

      Pages 155-163Open Access

   4. The Hurwitz order

      • John Voight

      Pages 165-179Open Access

   5. Ternary quadratic forms over local fields

      • John Voight

      Pages 181-199Open Access

   6. Quaternion algebras over local fields

      • John Voight

      Pages 201-215Open Access

   7. Quaternion algebras over global fields

      • John Voight

      Pages 217-240Open Access

   8. Discriminants

      • John Voight

      Pages 241-256Open Access

   9. Quaternion ideals and invertibility

      • John Voight

      Pages 257-276Open Access

   10. Classes of quaternion ideals

       • John Voight

       Pages 277-296Open Access

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.

• 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

“The book contains a huge amount of interesting and very well-chosen exercises. … This ‘encyclopedic’ character of the text may play an important role both as a guide to some special topics and as a source of information for both students and those whose research in related fields creates a need to familiarize themselves with the knowledge of the case when quaternion algebras are relevant.” (Juliusz Brzeziński, Mathematical Reviews, September, 2022)

• Department of Mathematics, Dartmouth College, Hanover, USA

  John Voight

John Voight is Professor of Mathematics at Dartmouth College in Hanover, New Hampshire. His research interests lie in arithmetic algebraic geometry and number theory, with a particular interest in computational aspects. He has taught graduate courses in algebra, number theory, cryptography, as well as the topic of this book, quaternion algebras.