Rational Quadratic FormsCourier Dover Publications, 08.08.2008 - 413 Seiten This exploration of quadratic forms over rational numbers and rational integers offers an excellent elementary introduction to many aspects of a classical subject, including recent developments. The author, a Professor Emeritus at Trinity College, University of Cambridge, offers a largely self-contained treatment that develops most of the prerequisites.Topics include the theory of quadratic forms over local fields, forms with integral coefficients, genera and spinor genera, reduction theory for definite forms, and Gauss' composition theory. The final chapter explains how to formulate the proofs in earlier chapters independently of Dirichlet's theorems related to the existence of primes in arithmetic progressions. Specialists will particularly value the several helpful appendixes on class numbers, Siegel's formulas, Tamagawa numbers, and other topics. Each chapter concludes with many exercises and hints, plus notes that include historical remarks and references to the literature. |
Inhalt
Introduction | 1 |
padic numbers | 8 |
Quadratic Forms over a Field | 11 |
pAdic Numbers | 34 |
Quadratic Forms Over Local Fields | 55 |
Tools from the Geometry of Numbers | 67 |
Forms over the rationals | 75 |
Forms over integral domains | 102 |
Spinor Genera | 196 |
The Reduction of Positive Definite Quadratic Forms | 255 |
Automorphs of Integral Forms | 284 |
Spinor 13 Automorphs of integral forms | 302 |
Composition of Binary Quadratic Forms | 331 |
genera | 332 |
Definite Forms | 362 |
Note Dotted lines indicate that only the material about binary | 371 |
Integral Forms over the Rational Integers | 127 |
Orthogonal 12 Reduction of definite forms | 131 |
The Spin and Orthogonal Groups | 169 |
Tamagawa Numbers | 379 |
Note on Determinants | 403 |
Andere Ausgaben - Alle anzeigen
Häufige Begriffe und Wortgruppen
a₁ a₂ algebraic number algebraic number fields anisotropic autometry ax² b₂ binary forms c₁ Chapter 9 clearly coefficients concludes the proof condition Corollary defined definite forms denote dimension Dirichlet's theorem e₁ elements equivalence class example f₁ finite set follows form f form f(x form of determinant fundamental discriminant Further given h₁ Hasse Principle Hence Hint homomorphism implies indefinite integral automorphs integrally equivalent isotropic lattice Let f Let f(x linear m₁ matrix modulo notation Note O+(f O+(V orthogonal orthogonal group Pell's equation positive definite positive integer primitive integral principal ideal domain proof of Theorem properly equivalent properties prove quadratic forms quadratic space rational regular quadratic s₁ satisfies Section Show Siegel solution Spin(V spinor genera spinor genus subgroup t₁ ternary form Theorem 3.1 theory u₁ u₂ v₁ variables Witt group x₁ Z-equivalent Z-lattice