算术教程
2009-8
世界图书出版公司
Jean-Pierre Serre
115
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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.
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.
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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当然还是数论的内容，写的比较现代。
zhende buvuo
书不错，价格有多高