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
pAdic Numbers | 34 |
Quadratic Forms Over Local Fields | 55 |
Tools from the Geometry of Numbers | 67 |
Quadratic Forms over the Rationals | 75 |
Quadratic Forms over Integral Domains | 102 |
Integral pAdic Forms | 111 |
Integral Forms over the Rational Integers | 127 |
The Spin and Orthogonal Groups | 169 |
The Reduction of Positive Definite Quadratic Forms | 255 |
Automorphs of Integral Forms | 284 |
Composition of Binary Quadratic Forms | 331 |
Definite Forms | 362 |
Tamagawa Numbers | 375 |
| 391 | |
| 405 | |
Spinor Genera | 196 |
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Häufige Begriffe und Wortgruppen
algebraic number fields anisotropic autometry basis binary forms Chapter 11 Chapter 9 classically integral form clearly coefficients concludes the proof Corollary corresponding defined denote dimension Dirichlet's theorem discriminant domain elements equivalence class example finite number finite set follows form f form f(x form of determinant formula fundamental discriminant Further Gauss given gives Hasse Principle Hence Hint homomorphism implies indefinite integral automorphs integral vector integrally equivalent isotropic isotropic over Q lattice Let f Let f(x linear matrix modular forms modulo Norm Residue Symbol notation Note orthogonal group p-adic unit Pell's equation positive integer precisely primitive integral proof of Theorem properly equivalent properties prove quadratic forms quadratic space rational reduced forms satisfies Section set of primes Show Siegel solution spin group Spin(V spinor genera spinor genus subgroup ternary form Theorem 3.1 theory unimodular variables Z-lattice Z,-equivalent