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.
Author(s): John Voight
Series: Graduate Texts in Mathematics, 288
Publisher: Springer
Year: 2021
Language: English
Pages: 908
City: Cham
Preface
Acknowledgements
Contents
1 Introduction
1.1 Hamilton's quaternions
1.2 Algebra after the quaternions
1.3 Quadratic forms and arithmetic
1.4 Modular forms and geometry
1.5 Conclusion
Exercises
Part I Algebra
2 Beginnings
2.1 Conventions
2.2 Quaternion algebras
2.3 Matrix representations
2.4 Rotations
Exercises
3 Involutions
3.1 Conjugation
3.2 Involutions
3.3 Reduced trace and reduced norm
3.4 Uniqueness and degree
3.5 Quaternion algebras
Exercises
4 Quadratic forms
4.1 Reduced norm as quadratic form
4.2 Basic definitions
4.3 Discriminants, nondegeneracy
4.4 Nondegenerate standard involutions
4.5 Special orthogonal groups
Exercises
5 Ternary quadratic forms and quaternion algebras
5.1 Reduced norm as quadratic form
5.2 Isomorphism classes of quaternion algebras
5.3 Clifford algebras
5.4 Splitting
5.5 Conics, embeddings
5.6 Orientations
Exercises
6 Characteristic 2
6.1 Separability
6.2 Quaternion algebras
6.3 Quadratic forms
6.4 Characterizing quaternion algebras
Exercises
7 Simple algebras
7.1 Motivation and summary
7.2 Simple modules
7.3 Wedderburn–Artin
7.4 Jacobson radical
7.5 Central simple algebras
7.6 Quaternion algebras
7.7 The Skolem–Noether theorem
7.8 Reduced trace and norm, universality
7.9 Separable algebras
Exercises
8 Simple algebras and involutions
8.1 The Brauer group and involutions
8.2 Biquaternion algebras
8.3 Brauer group
8.4 Positive involutions
8.5 Endomorphism algebras of abelian varieties
Exercises
Part II Arithmetic
9 Lattices and integral quadratic forms
9.1 Integral structures
9.2 Bits of commutative algebra
9.3 Lattices
9.4 Localizations
9.5 Completions
9.6 Index
9.7 Quadratic forms
9.8 Normalized form
Exercises
10 Orders
10.1 Lattices with multiplication
10.2 Orders
10.3 Integrality
10.4 Maximal orders
10.5 Orders in a matrix ring
Exercises
11 The Hurwitz order
11.1 The Hurwitz order
11.2 Hurwitz units
11.3 Euclidean algorithm
11.4 Unique factorization
11.5 Finite quaternionic unit groups
Exercises
12 Ternary quadratic forms over local fields
12.1 The p-adic numbers and local quaternion algebras
12.2 Local fields
12.3 Classification via quadratic forms
12.4 Hilbert symbol
Exercises
13 Quaternion algebras over local fields
13.1 Extending the valuation
13.2 Valuations
13.3 Classification via extensions of valuations
13.4 Consequences
13.5 Some topology
Exercises
14 Quaternion algebras over global fields
14.1 Ramification
14.2 Hilbert reciprocity over the rationals
14.3 Hasse–Minkowski theorem over the rationals
14.4 Global fields
14.5 Ramification and discriminant
14.6 Quaternion algebras over global fields
14.7 Theorems on norms
Exercises
15 Discriminants
15.1 Discriminantal notions
15.2 Discriminant
15.3 Quadratic forms
15.4 Reduced discriminant
15.5 Maximal orders and discriminants
15.6 Duality
Exercises
16 Quaternion ideals and invertibility
16.1 Quaternion ideals
16.2 Locally principal, compatible lattices
16.3 Reduced norms
16.4 Algebra and absolute norm
16.5 Invertible lattices
16.6 Invertibility with a standard involution
16.7 One-sided invertibility
16.8 Invertibility and the codifferent
Exercises
17 Classes of quaternion ideals
17.1 Ideal classes
17.2 Matrix ring
17.3 Classes of lattices
17.4 Types of orders
17.5 Finiteness of the class set: over the integers
17.6 Example
17.7 Finiteness of the class set: over number rings
17.8 Eichler's theorem
Exercises
18 Two-sided ideals and the Picard group
18.1 Noncommutative Dedekind domains
18.2 Prime ideals
18.3 Invertibility
18.4 Picard group
18.5 Classes of two-sided ideals
Exercises
19 Brandt groupoids
19.1 Composition laws and ideal multiplication
19.2 Example
19.3 Groupoid structure
19.4 Brandt groupoid
19.5 Brandt class groupoid
19.6 Quadratic forms
Exercises
20 Integral representation theory
20.1 Projectivity, invertibility, and representation theory
20.2 Projective modules
20.3 Projective modules and invertible lattices
20.4 Jacobson radical
20.5 Local Jacobson radical
20.6 Integral representation theory
20.7 Stable class group and cancellation
Exercises
21 Hereditary and extremal orders
21.1 Hereditary and extremal orders
21.2 Extremal orders
21.3 Explicit description of extremal orders
21.4 Hereditary orders
21.5 Classification of local hereditary orders
Exercises
22 Quaternion orders and ternary quadratic forms
22.1 Quaternion orders and ternary quadratic forms
22.2 Even Clifford algebras
22.3 Even Clifford algebra of a ternary quadratic module
22.4 Over a PID
22.5 Twisting and final bijection
Exercises
23 Quaternion orders
23.1 Highlights of quaternion orders
23.2 Maximal orders
23.3 Hereditary orders
23.4 Eichler orders
23.5 Bruhat–Tits tree
Exercises
24 Quaternion orders: second meeting
24.1 Advanced quaternion orders
24.2 Gorenstein orders
24.3 Eichler symbol
24.4 Chains of orders
24.5 Bass and basic orders
24.6 Tree of odd Bass orders
Exercises
Part III Analysis
25 The Eichler mass formula
25.1 Weighted class number formula
25.2 Imaginary quadratic class number formula
25.3 Eichler mass formula: over the rationals
25.4 Class number one and type number one
Exercises
26 Classical zeta functions
26.1 Eichler mass formula
26.2 Analytic class number formula
26.3 Classical zeta functions of quaternion algebras
26.4 Counting ideals in a maximal order
26.5 Eichler mass formula: maximal orders
26.6 Eichler mass formula: general case
26.7 Class number one
26.8 Functional equation and classification
Exercises
27 Adelic framework
27.1 The rational adele ring
27.2 The rational idele group
27.3 Rational quaternionic adeles and ideles
27.4 Adeles and ideles
27.5 Class field theory
27.6 Noncommutative adeles
27.7 Reduced norms
Exercises
28 Strong approximation
28.1 Beginnings
28.2 Strong approximation for SL2Q
28.3 Elementary matrices
28.4 Strong approximation and the ideal class set
28.5 Statement and first applications
28.6 Further applications
28.7 First proof
28.8 Second proof
28.9 Normalizer groups
28.10 Stable class group
Exercises
29 Idelic zeta functions
29.1 Poisson summation and the Riemann zeta function
29.2 Idelic zeta functions, after Tate
29.3 Measures
29.4 Modulus and Fourier inversion
29.5 Local measures and zeta functions: archimedean case
29.6 Local measures: commutative nonarchimedean case
29.7 Local zeta functions: nonarchimedean case
29.8 Idelic zeta functions
29.9 Convergence and residue
29.10 Main theorem
29.11 Tamagawa numbers
Exercises
30 Optimal embeddings
30.1 Representation numbers
30.2 Sums of three squares
30.3 Optimal embeddings
30.4 Counting embeddings, idelically: the trace formula
30.5 Local embedding numbers: maximal orders
30.6 Local embedding numbers: Eichler orders
30.7 Global embedding numbers
30.8 Class number formula
30.9 Type number formula
Exercises
31 Selectivity
31.1 Selective orders
31.2 Selectivity conditions
31.3 Selectivity setup
31.4 Outer selectivity inequalities
31.5 Middle selectivity equality
31.6 Optimal selectivity conclusion
31.7 Selectivity, without optimality
31.8 Isospectral, nonisometric manifolds
Exercises
Part IV Geometry and topology
32 Unit groups
32.1 Quaternion unit groups
32.2 Structure of units
32.3 Units in definite quaternion orders
32.4 Finite subgroups of quaternion unit groups
32.5 Cyclic subgroups
32.6 Dihedral subgroups
32.7 Exceptional subgroups
Exercises
33 Hyperbolic plane
33.1 The beginnings of hyperbolic geometry
33.2 Geodesic spaces
33.3 Upper half-plane
33.4 Classification of isometries
33.5 Geodesics
33.6 Hyperbolic area and the Gauss–Bonnet formula
33.7 Unit disc and Lorentz models
33.8 Riemannian geometry
Exercises
34 Discrete group actions
34.1 Topological group actions
34.2 Summary of results
34.3 Covering space and wandering actions
34.4 Hausdorff quotients and proper group actions
34.5 Proper actions on a locally compact space
34.6 Symmetric space model
34.7 Fuchsian groups
34.8 Riemann uniformization and orbifolds
Exercises
35 Classical modular group
35.1 The fundamental set
35.2 Binary quadratic forms
35.3 Moduli of lattices
35.4 Congruence subgroups
Exercises
36 Hyperbolic space
36.1 Hyperbolic space
36.2 Isometries
36.3 Unit ball, Lorentz, and symmetric space models
36.4 Bianchi groups and Kleinian groups
36.5 Hyperbolic volume
36.6 Picard modular group
Exercises
37 Fundamental domains
37.1 Dirichlet domains for Fuchsian groups
37.2 Ford domains
37.3 Generators and relations
37.4 Dirichlet domains
37.5 Hyperbolic Dirichlet domains
37.6 Poincaré's polyhedron theorem
37.7 Signature of a Fuchsian group
37.8 The (6, 4, 2)-triangle group
37.9 Unit group for discriminant 6
Exercises
38 Quaternionic arithmetic groups
38.1 Rational quaternion groups
38.2 Isometries from quaternionic groups
38.3 Discreteness
38.4 Compactness and finite generation
38.5 Arithmetic groups, more generally
38.6 Modular curves, seen idelically
38.7 Double cosets
Exercises
39 Volume formula
39.1 Statement
39.2 Volume setup
39.3 Volume derivation
39.4 Genus formula
Exercises
Part V Arithmetic geometry
40 Classical modular forms
40.1 Functions on lattices
40.2 Eisenstein series as modular forms
40.3 Classical modular forms
40.4 Theta series
40.5 Hecke operators
Exercises
41 Brandt matrices
41.1 Brandt matrices, neighbors, and modular forms
41.2 Brandt matrices
41.3 Commutativity of Brandt matrices
41.4 Semisimplicity
41.5 Eichler trace formula
Exercises
42 Supersingular elliptic curves
42.1 Supersingular elliptic curves
42.2 Supersingular isogenies
42.3 Equivalence of categories
42.4 Supersingular endomorphism rings
Exercises
43 QM abelian surfaces
43.1 QM abelian surfaces
43.2 QM by discriminant 6
43.3 Genus 2 curves
43.4 Complex abelian varieties
43.5 Complex abelian surfaces
43.6 Abelian surfaces with QM
43.7 Real points, CM points
43.8 Canonical models
43.9 Modular forms
Exercises
Symbol Definition List
Bibliography
Index
