A Course in Arithmetic (Graduate Texts in Mathematics v. 7 1st Corrected ed. 1973. Corr. 3rd printing 1996)
By: Jean-Pierre Serre (author)Hardback
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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.
I-Algebraic Methods.- I-Finite fields.- 1-Generalities.- 2-Equations over a finite field.- 3-Quadratic reciprocity law.- Appendix-Another proof of the quadratic reciprocity law.- II - p-adic fields.- 1-The ring Zp and the field Qp.- 2-p-adic equations.- 3-The multiplicative group of Qp.- III-Hilbert symbol.- 1-Local properties.- 2-Global properties.- IV-Quadratic forms over Qp and over Q.- 1-Quadratic forms.- 2-Quadratic forms over Qp.- 3-Quadratic forms over Q.- Appendix-Sums of three squares.- V-Integral quadratic forms with discriminant +- 1.- 1-Preliminaries.- 2-Statement of results.- 3-Proofs.- II-Analytic Methods.- 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.- VII-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.- Index of Definitions.- Index of Notations.
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- ID: 9780387900407
1st Corrected ed. 1973. Corr. 3rd printing 1996
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