Arithmetic Of Quadratic Forms

CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS, P. SARNAK, C. T. C. WALL 106 Arithmetic of quadratic forms YOSHIYUKI KITAOKA Department of Mathematics, Nagoya University Arithmetic of quadratic forms AMBRIDGE UNIVERSITY PRESS PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1993 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 First paperback edition (with corrections) 1999 Typeset in Computer Modern 10/12pt, in AMS-T [EPCJ A catalogue record for this book is available from the British Library ISBN 0 521 40475 4 hardback ISBN 0 521 64996 X paperback Transferred to digital printing 2003 Contents Preface vii Notation ix Chapter 1 General theory of quadratic forms 1.1 Symmetric bilinear forms 1.2 Quadratic forms 1.3 Quadratic forms over finite fields 1.4 The Clifford algebra 1.5 Quaternion algebras 1.6 The spinor norm 1.7 Scalar extensions 1 1 3 12 20 24 29 32 Chapter 2 Positive definite quadratic forms over R 2.1 Reduction theory 2.2 An estimate of Hermite's constant 33 33 42 Chapter 3 Quadratic forms over local fields 3.1 p-adic numbers 3.2 The quadratic residue symbol 3.3 The Hilbert symbol 3.4 The Hasse invariant 3.5 Classification of quadratic spaces over p-adic number fields 47 47 54 56 60 Chapter 4 Quadratic forms over Q 4.1 Quadratic forms over Q 64 64 52 Contents vi Chapter 5 Quadratic forms over the p-adic integer ring 5.1 Dual lattices 5.2 Maximal and modular lattices 5.3 Jordan decompositions 5.4 Extension theorems 5.5 The spinor norm 5.6 Local densities Chapter 6 Quadratic forms over Z 6.1 Fundamentals 6.2 Approximation theorems 6.3 Genus, spinor genus and class 6.4 Representation of codimension 1 6.5 Representation of codimension 2 6.6 Representation of codimension > 3 6.7 Orthogonal decomposition 6.8 The Minkowski-Siegel formula 70 70 71 79 86 92 94 129 129 134 147 151 157 164 169 173 Chapter 7 Some functorial properties of positive definite quadratic forms 189 7.1 Positive lattices of E-type 190 7.2 A fundamental lemma 199 7.3 Weighted graphs 217 7.4 The tensor product of positive lattices 222 7.5 Scalar extension of positive lattices 239 Notes References 250 Index 269 263 Preface The purpose of this book is to introduce the reader to the arithmetic of quadratic forms. Quadratic forms in this book are mainly considered over the rational number field or the ring of rational integers and their completions. It is of course possible to discuss quadratic forms over more general numbe