Shimura Arithmetic of Quadratic Forms

1. Auflage 2010
ISBN: 978-1-4419-1732-4
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 238 Seiten

Reihe: Mathematics and Statistics

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.

Autoren/Hrsg.

Shimura, Goro

Inhaltsverzeichnis

1;PREFACE;5
2;CONTENTS;8
3;NOTATION AND TERMINOLOGY;10
4;I THE QUADRATIC RECIPROCITY LAW;11
4.1;1. Elementary facts;11
4.2;2. Structure of (Z/mZ)×;14
4.3;3. The quadratic reciprocity law;15
4.4;4. Lattices in a vector space;21
4.5;5. Modules over a principal ideal domain;22
5;II ARITHMETIC IN AN ALGEBRAIC NUMBER FIELD;25
5.1;6. Valuations and p-adic numbers;25
5.2;7. Hensel’s lemma and its applications;32
5.3;8. Integral elements in algebraic extensions;35
5.4;9. Order functions in algebraic extensions;37
5.5;10. Ideal theory in an algebraic number field;45
6;III VARIOUS BASIC THEOREMS;56
6.1;11. The tensor product of fields;56
6.2;12. Units and the class number of a number field;59
6.3;13. Ideals in an extension of a number field;66
6.4;14. The discriminant and different;68
6.5;15. Adeles and ideles;75
6.6;16. Galois extensions;80
6.7;17. Cyclotomic fields;84
7;IV ALGEBRAS OVER A FIELD;88
7.1;18. Semisimple and simple algebras;88
7.2;19. Central simple algebras;95
7.3;20. Quaternion algebras;104
7.4;21. Arithmetic of semisimple algebras;109
8;V QUADRATIC FORMS;124
8.1;22. Algebraic theory of quadratic forms;124
8.2;23. Clifford algebras;129
8.3;24. Clifford groups and spin groups;136
8.4;25. Lower-dimensional cases;142
8.5;26. The Hilbert reciprocity law;149
8.6;27. The Hasse principle;152
9;VI DEEPER ARITHMETIC OF QUADRATIC FORMS;161
9.1;28. Classification of quadratic spaces over local and global fields;161
9.2;29. Lattices in a quadratic space;169
9.3;30. The genus and class of a lattice and a matrix;179
9.4;31. Integer-valued quadratic forms;187
9.5;32. Strong approximation in the indefinite case;194
9.6;33. Integer-valued symmetric forms;205
10;VII QUADRATIC DIOPHANTINE EQUATIONS;211
10.1;34. A historical perspective;211
10.2;35. Basic theorems of quadratic Diophantine equations;214
10.3;36. Classification of binary forms;221
10.4;37. New mass formulas;232
10.5;38. The theory of genera;236
11;REFERENCES;241
12;INDEX;243