Rational Quadratic Forms 0121632601
The material of the book is largely nineteenth century but the treatment is structured by two twentieth century insights
409 95 4MB
English Pages 431 Year 1978
Table of contents :
Cassels ,J. W. S.(F.R.S) Rational Quadratic Forms, L.M.S. vl.13(AP,1978)(ISBN 0121632601)(600dpi)(431p) ......Page 4
Copyright ......Page 5
Contents xiii ......Page 14
Preface v ......Page 6
Acknowledgements xi ......Page 12
Leitfaden xii ......Page 13
1.1 Introduction 1 ......Page 18
1.2 Basic Notions 5 ......Page 22
1.3 Prospect 8 ......Page 25
2.1 Introduction 11 ......Page 28
2.2 Isotropic Spaces 15 ......Page 32
2.3 Normal Bases 16 ......Page 33
2.4 Isometries and Autometries 18 ......Page 35
2.5 The Grothendieck and Witt Groups 22 ......Page 39
2.6 Singular Forms 27 ......Page 44
Examples 28 ......Page 45
3.1 Introduction 34 ......Page 51
3.2 Norm Residue Symbol 41 ......Page 58
3.3 Local and Global 44 ......Page 61
3.4 Hensel’s Lemma 47 ......Page 64
Notes 48 ......Page 65
Examples 49 ......Page 66
4.1 Introduction 55 ......Page 72
4.2 The Proofs 56 ......Page 73
4.3 The Witt Group 63 ......Page 80
Examples 66 ......Page 83
5.2 The Tools 67 ......Page 84
5.3 Background 72 ......Page 89
Examples 74 ......Page 91
6.1 Introduction 75 ......Page 92
6.2 The Weak Hasse Principle 77 ......Page 94
6.3 The Strong Hasse Principle, n < 2 78 ......Page 95
6.4 The Strong Hasse Principle, n = 3 78 ......Page 85
6.5 The Strong Hasse Principle, n = 4 83 ......Page 100
6.6 The Strong Hasse Principle, n > 5 84 ......Page 101
6.7 An Existence Theorem 85 ......Page 102
6.8 Size of Solutions 86 ......Page 103
6.9 An Approximation Theorem 89 ......Page 106
6.10 An Application: Finite Projective Planes 91 ......Page 108
6.11 The Witt Group 93 ......Page 110
Notes 96 ......Page 113
Examples 99 ......Page 116
7.2 Quadratic Forms and Lattices 102 ......Page 119
7.3 Lattices 104 ......Page 121
7.4 Singular Forms 108 ......Page 125
Notes 109 ......Page 126
Examples 110 ......Page 127
8.1 Introduction 111 ......Page 128
8.2 Bases of Z(p,n) 112 ......Page 129
8.3 Canonical Forms 113 ......Page 130
8.4 Canonical Forms, p = 2 117 ......Page 134
8.5 Approximation Theorems 123 ......Page 140
Examples 124 ......Page 141
9.1 Introduction 127 ......Page 144
9.2 Bases of Z^n 132 ......Page 149
9.3 The Finiteness Theorem 134 ......Page 151
9.4 Genera: Elementary Properties 139 ......Page 156
9.5 Existence of Genera: Representations 141 ......Page 158
9.6 Quantitative Study of Representations 144 ......Page 161
9.7 Semi-Equivalence 154 ......Page 171
9.8 Representation by Individual Forms 157 ......Page 174
Examples 161 ......Page 178
10.1 Introduction 169 ......Page 186
10.2 The Clifford Algebra 171 ......Page 188
10.3 The Spinor Norm and the Spin Group 175 ......Page 192
10.4 Lattices over Integral Domains 182 ......Page 199
10.5 Topological Considerations 184 ......Page 201
10.6 Change of Fields and Rings 185 ......Page 202
10.7 The Strong Approximation Theorem 186 ......Page 203
Notes 191 ......Page 208
Examples 192 ......Page 209
11.1 Introduction 196 ......Page 213
11.2 Localization of Lattices 204 ......Page 221
11.3 Number of Spinor Genera 207 ......Page 224
11.4 An Alternative Approach 215 ......Page 232
11.5 Simultaneous Bases of Two Lattices 221 ......Page 238
11.6 The Language of Forms 224 ......Page 241
11.7 Representation by Spinor Genera 227 ......Page 244
11.8 A Generalized Strong Approximation 230 ......Page 247
11.9 Representation by Definite Forms 235 ......Page 252
Notes 249 ......Page 266
Examples 251 ......Page 268
12.1 Introduction 255 ......Page 272
12.2 Successive Minima 260 ......Page 277
12.3 Reduced Forms and Siegel Domains 263 ......Page 280
12.4 Siegel Domains 266 ......Page 283
12.5 Geometry of Definite and Reduced Forms 269 ......Page 286
12.6 Geometry of the Binary Case 273 ......Page 290
12.7 Geometry of the General Case 277 ......Page 294
Notes 280 ......Page 297
Examples 282 ......Page 299
13.1 Introduction 284 ......Page 301
13.2 Hermite Reduction: Anisotropic Forms 285 ......Page 302
13.3 Binary Forms 289 ......Page 306
13.4 Construction of Automorphs 298 ......Page 315
13.5 Isotropic Ternary Forms 301 ......Page 318
13.6 Representation by Anisotropic Ternaries 303 ......Page 320
13.7 The Non-Educlidean Plane 309 ......Page 326
13.8 Proof of Theorem 6.1 313 ......Page 330
13.9 Quaternary Forms 317 ......Page 334
13.10 Real Automorphs. General Case 320 ......Page 337
13.11 Hermite Reduction. Isotropic Forms 321 ......Page 338
13.12 Effectiveness 324 ......Page 341
Examples 328 ......Page 345
14.1 Introduction 331 ......Page 348
14.2 Composition of Binary Forms 333 ......Page 350
14.3 Duplication and Genera 339 ......Page 356
14.4 Ambiguous Forms and Classes 341 ......Page 358
14.5 Existence Theorem 343 ......Page 360
14.6 The 2-Component of the Class-Group and Pell’s Equation 345 ......Page 362
14.7 Elimination of Dirichlet’s Theorem 353 ......Page 370
Notes 354 ......Page 371
Examples 358 ......Page 375
A.2 Orthogonal Decompositions 362 ......Page 379
A. 3 Class-Numbers of Genera and Spinor-Genera 364 ......Page 381
Examples 366 ......Page 383
B.l Introduction 368 ......Page 385
B.2 Binary Forms 370 ......Page 387
B.3 Siegel’s Formulae 374 ......Page 391
B.4 Tamagawa Numbers 379 ......Page 396
B.5 Modular Forms 382 ......Page 399
Notes 388 ......Page 405
Examples 389 ......Page 406
References 391 ......Page 408
Note on Determinants 403 ......Page 420
Index of Terminology 405 ......Page 422
Index of Notation 409 ......Page 426
cover......Page 1
Cassels ,J. W. S.(F.R.S) Rational Quadratic Forms, L.M.S. vl.13(AP,1978)(ISBN 0121632601)(600dpi)(431p) ......Page 4
Copyright ......Page 5
Contents xiii ......Page 14
Preface v ......Page 6
Acknowledgements xi ......Page 12
Leitfaden xii ......Page 13
1.1 Introduction 1 ......Page 18
1.2 Basic Notions 5 ......Page 22
1.3 Prospect 8 ......Page 25
2.1 Introduction 11 ......Page 28
2.2 Isotropic Spaces 15 ......Page 32
2.3 Normal Bases 16 ......Page 33
2.4 Isometries and Autometries 18 ......Page 35
2.5 The Grothendieck and Witt Groups 22 ......Page 39
2.6 Singular Forms 27 ......Page 44
Examples 28 ......Page 45
3.1 Introduction 34 ......Page 51
3.2 Norm Residue Symbol 41 ......Page 58
3.3 Local and Global 44 ......Page 61
3.4 Hensel’s Lemma 47 ......Page 64
Notes 48 ......Page 65
Examples 49 ......Page 66
4.1 Introduction 55 ......Page 72
4.2 The Proofs 56 ......Page 73
4.3 The Witt Group 63 ......Page 80
Examples 66 ......Page 83
5.2 The Tools 67 ......Page 84
5.3 Background 72 ......Page 89
Examples 74 ......Page 91
6.1 Introduction 75 ......Page 92
6.2 The Weak Hasse Principle 77 ......Page 94
6.3 The Strong Hasse Principle, n < 2 78 ......Page 95
6.4 The Strong Hasse Principle, n = 3 78 ......Page 85
6.5 The Strong Hasse Principle, n = 4 83 ......Page 100
6.6 The Strong Hasse Principle, n > 5 84 ......Page 101
6.7 An Existence Theorem 85 ......Page 102
6.8 Size of Solutions 86 ......Page 103
6.9 An Approximation Theorem 89 ......Page 106
6.10 An Application: Finite Projective Planes 91 ......Page 108
6.11 The Witt Group 93 ......Page 110
Notes 96 ......Page 113
Examples 99 ......Page 116
7.2 Quadratic Forms and Lattices 102 ......Page 119
7.3 Lattices 104 ......Page 121
7.4 Singular Forms 108 ......Page 125
Notes 109 ......Page 126
Examples 110 ......Page 127
8.1 Introduction 111 ......Page 128
8.2 Bases of Z(p,n) 112 ......Page 129
8.3 Canonical Forms 113 ......Page 130
8.4 Canonical Forms, p = 2 117 ......Page 134
8.5 Approximation Theorems 123 ......Page 140
Examples 124 ......Page 141
9.1 Introduction 127 ......Page 144
9.2 Bases of Z^n 132 ......Page 149
9.3 The Finiteness Theorem 134 ......Page 151
9.4 Genera: Elementary Properties 139 ......Page 156
9.5 Existence of Genera: Representations 141 ......Page 158
9.6 Quantitative Study of Representations 144 ......Page 161
9.7 Semi-Equivalence 154 ......Page 171
9.8 Representation by Individual Forms 157 ......Page 174
Examples 161 ......Page 178
10.1 Introduction 169 ......Page 186
10.2 The Clifford Algebra 171 ......Page 188
10.3 The Spinor Norm and the Spin Group 175 ......Page 192
10.4 Lattices over Integral Domains 182 ......Page 199
10.5 Topological Considerations 184 ......Page 201
10.6 Change of Fields and Rings 185 ......Page 202
10.7 The Strong Approximation Theorem 186 ......Page 203
Notes 191 ......Page 208
Examples 192 ......Page 209
11.1 Introduction 196 ......Page 213
11.2 Localization of Lattices 204 ......Page 221
11.3 Number of Spinor Genera 207 ......Page 224
11.4 An Alternative Approach 215 ......Page 232
11.5 Simultaneous Bases of Two Lattices 221 ......Page 238
11.6 The Language of Forms 224 ......Page 241
11.7 Representation by Spinor Genera 227 ......Page 244
11.8 A Generalized Strong Approximation 230 ......Page 247
11.9 Representation by Definite Forms 235 ......Page 252
Notes 249 ......Page 266
Examples 251 ......Page 268
12.1 Introduction 255 ......Page 272
12.2 Successive Minima 260 ......Page 277
12.3 Reduced Forms and Siegel Domains 263 ......Page 280
12.4 Siegel Domains 266 ......Page 283
12.5 Geometry of Definite and Reduced Forms 269 ......Page 286
12.6 Geometry of the Binary Case 273 ......Page 290
12.7 Geometry of the General Case 277 ......Page 294
Notes 280 ......Page 297
Examples 282 ......Page 299
13.1 Introduction 284 ......Page 301
13.2 Hermite Reduction: Anisotropic Forms 285 ......Page 302
13.3 Binary Forms 289 ......Page 306
13.4 Construction of Automorphs 298 ......Page 315
13.5 Isotropic Ternary Forms 301 ......Page 318
13.6 Representation by Anisotropic Ternaries 303 ......Page 320
13.7 The Non-Educlidean Plane 309 ......Page 326
13.8 Proof of Theorem 6.1 313 ......Page 330
13.9 Quaternary Forms 317 ......Page 334
13.10 Real Automorphs. General Case 320 ......Page 337
13.11 Hermite Reduction. Isotropic Forms 321 ......Page 338
13.12 Effectiveness 324 ......Page 341
Examples 328 ......Page 345
14.1 Introduction 331 ......Page 348
14.2 Composition of Binary Forms 333 ......Page 350
14.3 Duplication and Genera 339 ......Page 356
14.4 Ambiguous Forms and Classes 341 ......Page 358
14.5 Existence Theorem 343 ......Page 360
14.6 The 2-Component of the Class-Group and Pell’s Equation 345 ......Page 362
14.7 Elimination of Dirichlet’s Theorem 353 ......Page 370
Notes 354 ......Page 371
Examples 358 ......Page 375
A.2 Orthogonal Decompositions 362 ......Page 379
A. 3 Class-Numbers of Genera and Spinor-Genera 364 ......Page 381
Examples 366 ......Page 383
B.l Introduction 368 ......Page 385
B.2 Binary Forms 370 ......Page 387
B.3 Siegel’s Formulae 374 ......Page 391
B.4 Tamagawa Numbers 379 ......Page 396
B.5 Modular Forms 382 ......Page 399
Notes 388 ......Page 405
Examples 389 ......Page 406
References 391 ......Page 408
Note on Determinants 403 ......Page 420
Index of Terminology 405 ......Page 422
Index of Notation 409 ......Page 426
cover......Page 1
