\"本书讲述了：The first one is purely algebraic. Its objective is the classification ofquadratic forms over the field of rational numbers （Hasse-Minkowskitheorem）. It is achieved in Chapter IV. The first three chapters contain somepreliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols.Chapter V applies the preceding results to integral quadratic forms indiscriminant + 1. 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 arithmeticprogressions\"\" due to Dirichlet; this theorem is used at a critical point in thefirst part （Chapter 111, no. 2.2）. Chapter VII deals with modular forms,and in particular, with theta functions. Some of the quadratic forms ofChapter V reappear here.\"

Preface Part Ⅰ-Algebraic Methods ChapterI Finite fields 1-Generalities 2-Equations over a finite field 3-Quadratic reciprocity law Appendix-Another proof of the quadratic reciprocity law Chapter Ⅱ p-adic fields 1-The ring Zp and the field 2-p-adic equations 3-The multiplicative group of Chapter Ⅲ nHilbert symbol 1-Local properties 2-Global properties Chapter Ⅳ Quadratic forms over Qp and over Q 1-Quadratic forms 2-Quadratic forms over Q 3-Quadratic forms over Q Appendix Sums of three squares Chapter Ⅴ Integral quadratic forms with discriminant 1-Preliminaries 2-Statement of results 3-Proofs Part Ⅱ-Analytic Methods Chapter Ⅵ The theorem on arithmetic progressions 1-Characters of finite abelian groups 2-Dirichlet series 3-Zeta function and L functions 4-Density and Dirichlet theorem Chapter Ⅶ Modular forms 1-The modular group 2-Modular functions 3-The space of modular forms 4-Expansions at infinity 5-Hecke operators 6-Theta functions Bibliography Index of Definitions Index of Notations