Quaternion Algebras.

By: Voight, JohnMaterial type: TextTextSeries: Graduate Texts in Mathematics SeriesPublisher: Cham : Springer International Publishing AG, 2021Copyright date: �2021Edition: 1st edDescription: 1 online resource (877 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783030566944Genre/Form: Electronic books.Additional physical formats: Print version:: Quaternion AlgebrasLOC classification: QA251.5Online resources: Click to View
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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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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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2023. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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