Arithmetic of Quadratic Forms

by Yoshiyuki Kitaoka

• ISBN13: 9780521649964
• ISBN10: 052164996X
• Edition: Reprint
• Format: Paperback
• Copyright: 1999-05-28
• Publisher: Cambridge University Press

Summary

The aim of this book is to provide an introduction to quadratic forms that builds from basics up to the most recent results. Professor Kitaoka is well known for his work in this area, and in this book he covers many aspects of the subject, including lattice theory, Siegel's formula, and some results involving tensor products of positive definite quadratic forms. The reader is required to have only a knowledge of algebraic number fields, making this book ideal for graduate students and researchers wishing for an insight into quadratic forms.

Table of Contents

Preface	vii
Notation	ix
General theory of quadratic forms	1	(32)
Symmetric bilinear forms	1	(2)
Quadratic forms	3	(9)
Quadratic forms over finite fields	12	(8)
The Clifford algebra	20	(4)
Quaternion algebras	24	(5)
The spinor norm	29	(3)
Scalar extensions	32	(1)
Positive definite quadratic forms over R	33	(14)
Reduction theory	33	(9)
An estimate of Hermite's constant	42	(5)
Quadratic forms over local fields	47	(17)
p-adic numbers	47	(5)
The quadratic residue symbol	52	(2)
The Hilbert symbol	54	(2)
The Hasse invariant	56	(4)
Classification of quadratic spaces over p-adic number fields	60	(4)
Quadratic forms over Q	64	(6)
Quadratic forms over Q	64	(6)
Quadratic forms over the p-adic integer ring	70	(59)
Dual lattices	70	(1)
Maximal and modular lattices	71	(8)
Jordan decompositions	79	(7)
Extension theorems	86	(6)
The spinor norm	92	(2)
Local densities	94	(35)
Quadratic forms over Z	129	(60)
Fundamentals	129	(5)
Approximation theorems	134	(13)
Genus, spinor genus and class	147	(4)
Representation of codimension 1	151	(6)
Representation of codimension 2	157	(7)
Representation of codimension ≥ 3	164	(5)
Orthogonal decomposition	169	(4)
The Minkowski-Siegel formula	173	(16)
Some functorial properties of positive difinite Quadratic forms	189	(61)
Positive lattices of E-type	190	(9)
A fundamental lemma	199	(18)
Weighted graphs	217	(5)
The tensor product of positive lattices	222	(17)
Scalar extension of positive lattices	239	(11)
Notes	250	(13)
References	263	(6)
Index	269