算術(shù)教程

前言

　　This book is divided into two parts.　　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.　　The two parts correspond to lectures given in 1962 and 1964 to secondyear students at the Ecole Normale Superieure. A redaction of these lecturesin the form of duplicated notes， was made by J.-J. Saosuc （Chapters l-IV）and J.-P. Ramis and G. Ruget （Chapters VI-VIi）. They were very useful tome; I extend here my gratitude to their authors.

內(nèi)容概要

　　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.

書籍目錄

PrefacePart 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-ProofsPart 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 functionsBibliographyIndex of DefinitionsIndex of Notations

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