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《算术教程(英文版)》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 questi***: modular functi***,differential topology, finite groups. The second part （Chapters VI and VII） uses "***ytic" methods （holomor-phic functi***）. Chapter VI gives the proof of the "theorem on arithmeticprogressi***" 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 functi***. Some of the quadratic forms ofChapter V reappear here.

Preface

Part I-Algebraic Methods

ChapterI Finite fields

1-Generalities

2-Equati*** 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 equati***

3-The multiplicative group of

Chapter II 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 progressi***

1-Characters of finite abelian groups

2-Dirichlet series

3-Zeta function and L functi***

4-Density and Dirichlet theorem

Chapter Vll-Modular forms

1-The modular group

2-Modular functi***

3-The space of modular forms

4-Expansi*** at infinity

5-Hecke operators

6-Theta functi***

Bibliography

Index of Definiti***

Index of Notati***

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书籍介绍

《算术教程(英文版)》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 questi***: modular functi***,differential topology, finite groups. The second part （Chapters VI and VII） uses "***ytic" methods （holomor-phic functi***）. Chapter VI gives the proof of the "theorem on arithmeticprogressi***" 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 functi***. Some of the quadratic forms ofChapter V reappear here.