Arithmetic of Quadratic Forms

by Shimura, Goro

• ISBN13: 9781441917317
• ISBN10: 1441917314
• Edition: 1st
• Format: Hardcover
• Copyright: 2010-07-30
• Publisher: Springer Verlag

Summary

This book is divided in two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case.To make the book concise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are stated with an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.

Table of Contents

Preface	p. v
Notation and terminology	p. xi
The quadratic reciprocity law	p. 1
Elementary facts	p. 1
Structure of (Z/mZ)x	p. 4
The quadratic reciprocity law	p. 5
Lattices in a vector space	p. 11
Modules over a principal ideal domain	p. 12
Arithmetic in an algebraic number field	p. 15
Valuations and p-adic numbers	p. 15
Hensel's lemma and its applications	p. 22
Integral elements in algebraic extensions	p. 25
Order functions in algebraic extensions	p. 27
Ideal theory in an algebraic number field	p. 35
Various basic theorems	p. 47
The tensor product of fields	p. 47
Units and the class number of a number field	p. 50
Ideals in an extension of a number field	p. 57
The discriminant and different	p. 59
Adeles and ideles	p. 66
Galois extensions	p. 71
Cyclotomic fields	p. 75
Algebras over a field	p. 79
Semisimple and simple algebras	p. 79
Central simple algebras	p. 86
Quaternion algebras	p. 95
Arithmetic of semisimple algebras	p. 100
Quadratic forms over a field	p. 115
Algebraic theory of quadratic forms	p. 115
Clifford algebras	p. 120
Clifford groups and spin groups	p. 127
Lower-dimensional cases	p. 133
The Hilbert reciprocity law	p. 140
The Hasse principle	p. 143
Deeper arithmetic of quadratic forms	p. 153
Classification of quadratic forms over local and global fields	p. 153
Lattices in a quadratic space	p. 161
The genus and class of a lattice and a matrix	p. 171
Integer-valued quadratic forms	p. 179
Strong approximation in the indefinite case	p. 186
Integer-valued symmetric forms	p. 197
Quadratic Diophantine equations	p. 203
A historical perspective	p. 203
Basic theorems of quadratic Diophantine equations	p. 206
Classification of binary forms	p. 213
New mass formulas	p. 224
The theory of genera	p. 228
References	p. 233
Index	p. 235
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