Full Description
This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant +/- I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors.
Table of Contents
Preface v
Part I--Algebraic Methods
Finite fields 3 (8)
Generalities 3 (2)
Equations over a finite field 5 (1)
Quadratic reciprocity law 6 (5)
Another proof of the quadratic 9 (2)
reciprocity law
p-adic fields 11 (8)
The ring Zp and the field Qp 11 (2)
p-adic equations 13 (2)
The multiplicative group of Qp 15 (4)
Hilbert symbol 19 (8)
Local properties 19 (4)
Global properties 23 (4)
Quadratic forms over Qp and over Q 27 (21)
Quadratic forms 27 (8)
Quadratic forms over Qp 35 (6)
Quadratic forms over Q 41 (7)
Sums of three squares 45 (3)
Integral quadratic forms with discriminant 48 (13)
± 1
Preliminaries 48 (4)
Statement of results 52 (3)
Proofs 55 (6)
Part II---Analytic Methods
The theorem on arithmetic progressions 61 (16)
Characters of finite abelian groups 61 (3)
Dirichlet series 64 (4)
Zeta function and L functions 68 (5)
Density and Dirichlet theorem 73 (4)
Modular forms 77 (35)
The modular group 77 (2)
Modular functions 79 (5)
The space of modular forms 84 (6)
Expansions at infinity 90 (8)
Hecke operators 98 (8)
Theta functions 106(6)
Bibliography 112(2)
Index of Definitions 114(1)
Index of Notations 115
