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El. knyga: Quaternion Algebras

  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 288
  • Išleidimo metai: 28-Jun-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030566944
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Quaternion AlgebrasDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Graduate Texts in Mathematics 288
  • Išleidimo metai: 28-Jun-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030566944
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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 unitgroups 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 andmotivation are recapped throughout.

Recenzijos

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 Brzeziski, Mathematical Reviews, September, 2022)

1 Introduction
1(20)
1.1 Hamilton's quaternions
1(5)
1.2 Algebra after the quaternions
6(4)
1.3 Quadratic forms and arithmetic
10(1)
1.4 Modular forms and geometry
11(4)
1.5 Conclusion
15(1)
Exercises
16(5)
I Algebra
2 Beginnings
21(14)
2.1 Conventions
21(1)
2.2 Quaternion algebras
22(2)
2.3 Matrix representations
24(2)
2.4 Rotations
26(4)
Exercises
30(5)
3 Involutions
35(12)
3.1 Conjugation
35(1)
3.2 Involutions
36(2)
3.3 Reduced trace and reduced norm
38(1)
3.4 Uniqueness and degree
39(1)
3.5 Quaternion algebras
40(3)
Exercises
43(4)
4 Quadratic forms
47(18)
4.1 Reduced norm as quadratic form
47(1)
4.2 Basic definitions
48(4)
4.3 Discriminants, nondegeneracy
52(2)
4.4 Nondegenerate standard involutions
54(1)
4.5 Special orthogonal groups
55(4)
Exercises
59(6)
5 Ternary quadratic forms and quaternion algebras
65(20)
5.1 Reduced norm as quadratic form
65(1)
5.2 Isomorphism classes of quaternion algebras
66(3)
5.3 Clifford algebras
69(4)
5.4 Splitting
73(3)
5.5 Conics, embeddings
76(1)
5.6 Orientations
77(4)
Exercises
81(4)
6 Characteristic 2
85(10)
6.1 Separability
85(1)
6.2 Quaternion algebras
86(1)
6.3 Quadratic forms
87(2)
6.4 Characterizing quaternion algebras
89(4)
Exercises
93(2)
7 Simple algebras
95(28)
7.1 Motivation and summary
95(2)
7.2 Simple modules
97(4)
7.3 Semisimple modules and the Wedderburn-Artin theorem
101(2)
7.4 Jacobson radical
103(2)
7.5 Central simple algebras
105(2)
7.6 Quaternion algebras
107(1)
7.7 The Skolem-Noether theorem
108(4)
7.8 Reduced trace and norm, universality
112(3)
7.9 Separable algebras
115(2)
Exercises
117(6)
8 Simple algebras and involutions
123(16)
8.1 The Brauer group and involutions
123(1)
8.2 Biquaternion algebras
124(2)
8.3 Brauer group
126(2)
8.4 Positive involutions
128(3)
8.5 Endomorphism algebras of abelian varieties
131(2)
Exercises
133(6)
II Arithmetic
9 Lattices and integral quadratic forms
139(16)
9.1 Integral structures
139(1)
9.2 Bits of commutative algebra
140(1)
9.3 Lattices
141(2)
9.4 Localizations
143(2)
9.5 Completions
145(2)
9.6 Index
147(1)
9.7 Quadratic forms
148(2)
9.8 Normalized form
150(2)
Exercises
152(3)
10 Orders
155(10)
10.1 Lattices with multiplication
155(1)
10.2 Orders
156(2)
10.3 Integrality
158(1)
10.4 Maximal orders
159(1)
10.5 Orders in a matrix ring
160(1)
Exercises
161(4)
11 The Hurwitz order
165(16)
11.1 The Hurwitz order
165(1)
11.2 Hurwitz units
166(3)
11.3 Euclidean algorithm
169(1)
11.4 Unique factorization
170(2)
11.5 Finite quaternionic unit groups
172(4)
Exercises
176(5)
12 Ternary quadratic forms over local fields
181(20)
12.1 The p-adic numbers and local quaternion algebras
181(3)
12.2 Local fields
184(5)
12.3 Classification via quadratic forms
189(4)
12.4 Hilbert symbol
193(3)
Exercises
196(5)
13 Quaternion algebras over local fields
201(16)
13.1 Extending the valuation
201(1)
13.2 Valuations
202(3)
13.3 Classification via extensions of valuations
205(3)
13.4 Consequences
208(2)
13.5 Some topology
210(2)
Exercises
212(5)
14 Quaternion algebras over global fields
217(24)
14.1 Ramification
217(2)
14.2 Hilbert reciprocity over the rationals
219(4)
14.3 Hasse-Minkowski theorem over the rationals
223(4)
14.4 Global fields
227(3)
14.5 Ramification and discriminant
230(1)
14.6 Quaternion algebras over global fields
231(2)
14.7 Theorems on norms
233(2)
Exercises
235(6)
15 Discriminants
241(16)
15.1 Discriminantal notions
241(1)
15.2 Discriminant
242(3)
15.3 Quadratic forms
245(1)
15.4 Reduced discriminant
246(2)
15.5 Maximal orders and discriminants
248(1)
15.6 Duality
249(4)
Exercises
253(4)
16 Quaternion ideals and invertibility
257(20)
16.1 Quaternion ideals
257(2)
16.2 Locally principal, compatible lattices
259(2)
16.3 Reduced norms
261(2)
16.4 Algebra and absolute norm
263(2)
16.5 Invertible lattices
265(3)
16.6 Invertibility with a standard involution
268(3)
16.7 One-sided invertibility
271(2)
16.8 Invertibility and the codifferent
273(1)
Exercises
274(3)
17 Classes of quaternion ideals
277(20)
17.1 Ideal classes
277(2)
17.2 Matrix ring
279(1)
17.3 Classes of lattices
280(1)
17.4 Types of orders
281(3)
17.5 Finiteness of the class set: over the integers
284(1)
17.6 Example
285(2)
17.7 Finiteness of the class set: over number rings
287(4)
17.8 Eichler's theorem
291(2)
Exercises
293(4)
18 Two-sided ideals and the Picard group
297(14)
18.1 Noncommutative Dedekind domains
297(2)
18.2 Prime ideals
299(1)
18.3 Invertibility
300(3)
18.4 Picard group
303(2)
18.5 Classes of two-sided ideals
305(2)
Exercises
307(4)
19 Brandt groupoids
311(14)
19.1 Composition laws and ideal multiplication
311(3)
19.2 Example
314(1)
19.3 Groupoid structure
315(3)
19.4 Brandt groupoid
318(1)
19.5 Brandt class groupoid
319(2)
19.6 Quadratic forms
321(1)
Exercises
322(3)
20 Integral representation theory
325(18)
20.1 Projectivity, invertibility, and representation theory
325(2)
20.2 Projective modules
327(1)
20.3 Projective modules and invertible lattices
328(2)
20.4 Jacobson radical
330(2)
20.5 Local Jacobson radical
332(1)
20.6 Integral representation theory
333(2)
20.7 Stable class group and cancellation
335(5)
Exercises
340(3)
21 Hereditary and extremal orders
343(10)
21.1 Hereditary and extremal orders
343(1)
21.2 Extremal orders
344(2)
21.3 Explicit description of extremal orders
346(2)
21.4 Hereditary orders
348(2)
21.5 Classification of local hereditary orders
350(1)
Exercises
351(2)
22 Quaternion orders and ternary quadratic forms
353(22)
22.1 Quaternion orders and ternary quadratic forms
353(3)
22.2 Even Clifford algebras
356(3)
22.3 Even Clifford algebra of a ternary quadratic module
359(5)
22.4 Over a PID
364(4)
22.5 Twisting and final bijection
368(3)
Exercises
371(4)
23 Quaternion orders
375(18)
23.1 Highlights of quaternion orders
375(2)
23.2 Maximal orders
377(1)
23.3 Hereditary orders
378(4)
23.4 Eichler orders
382(4)
23.5 Bruhat-Tits tree
386(3)
Exercises
389(4)
24 Quaternion orders: second meeting
393(22)
24.1 Advanced quaternion orders
393(1)
24.2 Gorenstein orders
394(6)
24.3 Eichler symbol
400(3)
24.4 Chains of orders
403(3)
24.5 Bass and basic orders
406(3)
24.6 Tree of odd Bass orders
409(1)
Exercises
410(5)
III Analysis
25 The Eichler mass formula
415(16)
25.1 Weighted class number formula
415(1)
25.2 Imaginary quadratic class number formula
416(4)
25.3 Eichler mass formula: over the nationals
420(3)
25.4 Class number one and type number one
423(2)
Exercises
425(6)
26 Classical zeta functions
431(26)
26.1 Eichler mass formula
431(2)
26.2 Analytic class number formula
433(3)
26.3 Classical zeta functions of quaternion algebras
436(2)
26.4 Counting ideals in a maximal order
438(3)
26.5 Eichler mass formula: maximal orders
441(3)
26.6 Eichler mass formula: general case
444(3)
26.7 Class number one
447(1)
26.8 Functional equation and classification
447(5)
Exercises
452(5)
27 Adelic framework
457(20)
27.1 The rational adele ring
457(3)
27.2 The rational idele group
460(2)
27.3 Rational quaternionic adeles and ideles
462(1)
27.4 Adeles and ideles
463(2)
27.5 Class field theory
465(3)
27.6 Noncommutative adeles
468(4)
27.7 Reduced norms
472(1)
Exercises
473(4)
28 Strong approximation
477(26)
28.1 Beginnings
477(2)
28.2 Strong approximation for SL2(Q)
479(3)
28.3 Elementary matrices
482(1)
28.4 Strong approximation and the ideal class set
483(1)
28.5 Statement and first applications
484(3)
28.6 Further applications
487(3)
28.7 First proof
490(3)
28.8 Second proof
493(2)
28.9 Normalizer groups
495(3)
28.10 Stable class group
498(1)
Exercises
499(4)
29 Idelic zeta functions
503(38)
29.1 Poisson summation and the Riemann zeta function
503(2)
29.2 Idelic zeta functions, after Tate
505(2)
29.3 Measures
507(2)
29.4 Modulus and Fourier inversion
509(2)
29.5 Local measures and zeta functions: archimedean case
511(3)
29.6 Local measures: commutative nonarchimedean case
514(3)
29.7 Local zeta functions: nonarchimedean case
517(4)
29.8 Idelic zeta functions
521(4)
29.9 Convergence and residue
525(2)
29.10 Main theorem
527(8)
29.11 Tamagawa numbers
535(2)
Exercises
537(4)
30 Optimal embeddings
541(28)
30.1 Representation numbers
541(2)
30.2 Sums of three squares
543(2)
30.3 Optimal embeddings
545(2)
30.4 Counting embeddings, idelically: the trace formula
547(4)
30.5 Local embedding numbers: maximal orders
551(3)
30.6 Local embedding numbers: Eichler orders
554(5)
30.7 Global embedding numbers
559(1)
30.8 Class number formula
560(3)
30.9 Type number formula
563(2)
Exercises
565(4)
31 Selectivity
569(22)
31.1 Selective orders
569(3)
31.2 Selectivity conditions
572(1)
31.3 Selectivity setup
573(3)
31.4 Outer selectivity inequalities
576(1)
31.5 Middle selectivity equality
577(2)
31.6 Optimal selectivity conclusion
579(1)
31.7 Selectivity, without optimality
580(2)
31.8 Isospectral, nonisometric manifolds
582(3)
Exercises
585(6)
IV Geometry and topology
32 Unit groups
591(14)
32.1 Quaternion unit groups
591(1)
32.2 Structure of units
592(2)
32.3 Units in definite quatemion orders
594(2)
32.4 Finite subgroups of quatemion unit groups
596(1)
32.5 Cyclic subgroups
597(2)
32.6 Dihedral subgroups
599(2)
32.7 Exceptional subgroups
601(1)
Exercises
602(3)
33 Hyperbolic plane
605(24)
33.1 The beginnings of hyperbolic geometry
605(1)
33.2 Geodesic spaces
606(2)
33.3 Upper half-plane
608(3)
33.4 Classification of isometries
611(3)
33.5 Geodesics
614(2)
33.6 Hyperbolic area and the Gauss-Bonnet formula
616(2)
33.7 Unit disc and Lorentz models
618(3)
33.8 Riemannian geometry
621(3)
Exercises
624(5)
34 Discrete group actions
629(20)
34.1 Topological group actions
629(3)
34.2 Summary of results
632(1)
34.3 Covering space and wandering actions
632(2)
34.4 Hausdorff quotients and proper group actions
634(3)
34.5 Proper actions on a locally compact space
637(2)
34.6 Symmetric space model
639(1)
34.7 Fuchsian groups
640(2)
34.8 Riemann uniformization and orbifolds
642(3)
Exercises
645(4)
35 Classical modular group
649(14)
35.1 The fundamental set
649(6)
35.2 Binary quadratic forms
655(2)
35.3 Moduli of lattices
657(1)
35.4 Congruence subgroups
658(2)
Exercises
660(3)
36 Hyperbolic space
663(22)
36.1 Hyperbolic space
663(2)
36.2 Isometries
665(4)
36.3 Unit ball, Lorentz, and symmetric space models
669(2)
36.4 Bianchi groups and Kleinian groups
671(1)
36.5 Hyperbolic volume
672(4)
36.6 Picard modular group
676(4)
Exercises
680(5)
37 Fundamental domains
685(30)
37.1 Dirichlet domains for Fuchsian groups
685(3)
37.2 Ford domains
688(2)
37.3 Generators and relations
690(5)
37.4 Dirichlet domains
695(4)
37.5 Hyperbolic Dirichlet domains
699(1)
37.6 Poincare's polyhedron theorem
700(3)
37.7 Signature of a Fuchsian group
703(1)
37.8 The (6, 4, 2)-triangle group
704(3)
37.9 Unit group for discriminant 6
707(5)
Exercises
712(3)
38 Quaternionic arithmetic groups
715(16)
38.1 Rational quaternion groups
715(2)
38.2 Isometries from quaternionic groups
717(2)
38.3 Discreteness
719(2)
38.4 Compactness and finite generation
721(2)
38.5 Arithmetic groups, more generally
723(1)
38.6 Modular curves, seen idelically
724(2)
38.7 Double cosets
726(3)
Exercises
729(2)
39 Volume formula
731(14)
39.1 Statement
731(3)
39.2 Volume setup
734(2)
39.3 Volume derivation
736(1)
39.4 Genus formula
737(4)
Exercises
741(4)
V Arithmetic geometry
40 Classical modular forms
745(18)
40.1 Functions on lattices
745(4)
40.2 Eisenstein series as modular forms
749(3)
40.3 Classical modular forms
752(3)
40.4 Theta series
755(1)
40.5 Hecke operators
756(2)
Exercises
758(5)
41 Brandt matrices
763(20)
41.1 Brandt matrices, neighbors, and modular forms
763(4)
41.2 Brandt matrices
767(3)
41.3 Commutativity of Brandt matrices
770(3)
41.4 Semisimplicity
773(2)
41.5 Eichler trace formula
775(4)
Exercises
779(4)
42 Supersingular elliptic curves
783(16)
42.1 Supersingular elliptic curves
783(4)
42.2 Supersingular isogenies
787(4)
42.3 Equivalence of categories
791(2)
42.4 Supersingular endomorphism rings
793(3)
Exercises
796(3)
43 QM abelian surfaces
799(32)
43.1 QM abelian surfaces
799(3)
43.2 QM by discriminant 6
802(4)
43.3 Genus 2 curves
806(3)
43.4 Complex abelian varieties
809(5)
43.5 Complex abelian surfaces
814(3)
43.6 Abelian surfaces with QM
817(6)
43.7 Real points, CM points
823(1)
43.8 Canonical models
824(2)
43.9 Modular forms
826(2)
Exercises
828(3)
Symbol Definition List 831(8)
Bibliography 839(34)
Index 873
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.
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