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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 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 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 progressions

1-Characters of finite abelian groups

2-Dirichlet series

3-Zeta function and L functions

4-Density and Dirichlet theorem

Chapter Vll-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

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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 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.