A Course in Arithmetic

• Jean-Pierre Serre
Book

Part of the Graduate Texts in Mathematics book series (GTM, volume 7)

1. Front Matter
Pages i-ix
2. Algebraic Methods

1. Front Matter
Pages 1-1
2. Jean-Pierre Serre
Pages 3-10
3. Jean-Pierre Serre
Pages 11-18
4. Jean-Pierre Serre
Pages 19-26
5. Jean-Pierre Serre
Pages 27-47
6. Jean-Pierre Serre
Pages 48-58
3. Analytic Methods

1. Front Matter
Pages 59-59
2. Jean-Pierre Serre
Pages 61-76
3. Jean-Pierre Serre
Pages 77-111
4. Back Matter
Pages 112-119

Introduction

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.

Keywords

Arithmetik algebra arithmetic field finite field finite group modular form quadratic form ring zeta function

Authors and affiliations

• Jean-Pierre Serre
• 1
1. 1.Collège de FranceParis Cedex 05France

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4684-9884-4
• Copyright Information Springer-Verlag New York 1973
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-90041-4
• Online ISBN 978-1-4684-9884-4
• Series Print ISSN 0072-5285
• Series Online ISSN 2197-5612