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Quaternion Algebras.

Bibliographic Details
Main Author: Voight, John.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2021.
Edition:1st ed.
Series:Graduate Texts in Mathematics Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • 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.

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