Arithmetic of Quadratic Forms (eBook)

Goro Shimura (Autor)

eBook Download: PDF

2010 | 1. Auflage
238 Seiten
Springer-Verlag
978-1-4419-1732-4 (ISBN)

Infos

This book can be divided into two parts. The ?rst part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. The raison d’ˆ etre of the book is in the second part, and so let us ?rst explain the contents of the second part. There are two principal topics: (A) Classi?cation of quadratic forms; (B) Quadratic Diophantine equations. Topic (A) can be further divided into two types of theories: (a1) Classi?cation over an algebraic number ?eld; (a2) Classi?cation over the ring of algebraic integers. To classify a quadratic form ? over an algebraic number ?eld F, almost all previous authors followed the methods of Helmut Hasse. Namely, one ?rst takes ? in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely it was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to ? at each prime spot, which is the same as Eichler’s, but we de?ne it in a di?erent and more direct way, using Cli?ord algebras. In Sections 27 and 28 we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number ?eld, which is included in class ?eld theory. We prove it when the base ?eld is the rational number ?eld to make the book self-contained in that case.

Inhalt

PREFACE 5
CONTENTS 8
NOTATION AND TERMINOLOGY 10
I THE QUADRATIC RECIPROCITY LAW 11
1. Elementary facts 11
2. Structure of (Z/mZ)× 14
3. The quadratic reciprocity law 15
4. Lattices in a vector space 21
5. Modules over a principal ideal domain 22
II ARITHMETIC IN AN ALGEBRAIC NUMBER FIELD 25
6. Valuations and p-adic numbers 25
7. Hensel’s lemma and its applications 32
8. Integral elements in algebraic extensions 35
9. Order functions in algebraic extensions 37
10. Ideal theory in an algebraic number field 45
III VARIOUS BASIC THEOREMS 56
11. The tensor product of fields 56
12. Units and the class number of a number field 59
13. Ideals in an extension of a number field 66
14. The discriminant and different 68
15. Adeles and ideles 75
16. Galois extensions 80
17. Cyclotomic fields 84
IV ALGEBRAS OVER A FIELD 88
18. Semisimple and simple algebras 88
19. Central simple algebras 95
20. Quaternion algebras 104
21. Arithmetic of semisimple algebras 109
V QUADRATIC FORMS 124
22. Algebraic theory of quadratic forms 124
23. Clifford algebras 129
24. Clifford groups and spin groups 136
25. Lower-dimensional cases 142
26. The Hilbert reciprocity law 149
27. The Hasse principle 152
VI DEEPER ARITHMETIC OF QUADRATIC FORMS 161
28. Classification of quadratic spaces over local and global fields 161
29. Lattices in a quadratic space 169
30. The genus and class of a lattice and a matrix 179
31. Integer-valued quadratic forms 187
32. Strong approximation in the indefinite case 194
33. Integer-valued symmetric forms 205
VII QUADRATIC DIOPHANTINE EQUATIONS 211
34. A historical perspective 211
35. Basic theorems of quadratic Diophantine equations 214
36. Classification of binary forms 221
37. New mass formulas 232
38. The theory of genera 236
REFERENCES 241
INDEX 243

Produkt-Details

• Erscheint lt. Verlag: 9.8.2010
• Reihe/Serie: Springer Monographs in Mathematics
• Reihe/Serie: Springer Monographs in Mathematics
• Zusatzinfo: XII, 238 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Algebra • Clifford Algebras • Number Theory • Quadratic Diophantine equations • quadratic forms • Quadratic reciprocity law • Semisimple algebras • Strong approximation • Sums of squares
• ISBN-10: 1-4419-1732-2 / 1441917322
• ISBN-13: 978-1-4419-1732-4 / 9781441917324