Rational Quadratic Forms (London Mathematical Society Monograph)

The material of the book is largely nineteenth century but the treatment is structured by two twentieth century insights. The first, which seems to have come to its full recognition in the work of Hasse and Witt, is that the theory of forms over fields is logically simpler and more complete than that over rings. It is therefore appropriate, contrary to what seemed natural earlier, to study forms with rational coefficients and under rational equivalence before attacking integral forms and integral equivalence. The second major insight, due to Hensel and Hasse, is the perspective introduced by the p-adic view-point. This unveils the majestic simplicity of the logical structure: in particular it banishes for ever the need for the plethora of multifarious “characters” and “invariants” which earlier (and some later) authors use to distinguish forms which are p-adically inequivalent. The p-adic numbers are as natural as the reals—indeed it can plausibly be argued that they are logically simpler and that it is only by indoctrination that we feel that the reals are more familiar. However, as the p-adic numbers are not yet as well- known as they should be to the broad audience to which this book is addressed, no knowledge of them has been presupposed. 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 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