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.

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