橢圓曲線算術中的高等論題(英文)（簡體書）

《橢圓曲線算術中的高等論題(英文版)》內容簡介：美國哈佛大學從1977年開始，曾多次舉辦”橢圓曲線” 班，《橢圓曲線算術中的高等論題(英文版)》作者是該討論班成員之一。橢圓曲線是一個古老的數學課題，最近由于代數數論和代數幾何等現代數學的進展，使它得到了新的活力。《橢圓曲線算術中的高等論題(英文版)》是以1986年版的《橢圓曲線的算術理論》為藍本，但在知識體系上做了較大的改動形成了這不教程，講述上也更加專業，但在思想上是作者前《橢圓曲線算術中的高等論題(英文版)》的延續。包括橢圓和模型函數；復乘方法；橢圓曲線；Néron模型；復域上的橢圓曲線等內容。每章末都配有大量習題。目次：橢圓和模型函數；復乘方法；橢圓曲線；Néron模型；復域上的橢圓曲線。
讀者對象：適合數學專業的研究生和相關的科研人員。

Preface
Computer Packages
Acknowledgments
Introduction

CHAPTER Ⅰ
Elliptic and Modular Functions
The Modular Group
The Modular Curve X(1)
Modular Functions
Uniformization and Fields of Moduli
Elliptic Functions Revisited
q-Expansions of Elliptic Functions
q-Expansions of Modular Functions
Jacobis Product Formula for A(T)
Hecke Operators
Hecke Operators Acting on Modular Forms
L-Series Attached to Modular Forms
Exercises

CHAPTER Ⅱ
Complex Multiplication
Complex Multiplication over C
Rationality Questions
Class Field Theory —— A Brief Review
The Hilbert Class Field
The Maximal Abelian Extension
Integrality of j
Cyclotomic Class Field Theory
The Main Theorem of Complex Multiplication
The Associated GrSssencharacter
The L-Series Attached to a CM Elliptic Curve
Exercises

CHAPTER Ⅲ
Elliptic Surfaces
Elliptic Curves over Function Fields
The Weak Mordell-Weil Theorem
Elliptic Surfaces
Heights on Elliptic Curves over Unction Fields
Split Elliptic Surfaces and Sets of Bounded Height
The Mordell-Weil Theorem for Fhnction Fields
The Geometry of Algebraic Surfaces
The Geometry of Fibered Surfaces
The Geometry of Elliptic Surfaces
Heights and Divisors on Varieties
Specialization Theorems for Elliptic Surfaces
Integral Points on Elliptic Curves over Function Fields
Exercises

CHAPTER Ⅳ
The N6ron Model
Group Varieties
Schemes and S-Schemes
Group Schemes
Arithmetic Surfaces
N6ron Models
Existence of N6ron Models
Intersection Theory, Minimal Models, and Blowing-Up
The Special Fiber of a N6ron Model
Tates Algorithm to Compute the Special Fiber
The Conductor of an Elliptic Curve
Oggs Formula
Exercises

CHAPTER Ⅴ
Elliptic Curves over Complete Fields
Elliptic Curves over C
Elliptic Curves over R
The Tate Curve
The Tate Map Is Surjective
Elliptic Curves over p-adic Fields
Some Applications of p-adic Uniformization
Exercises

CHAPTER Ⅵ
Local Height Functions
Existence of Local Height Functions
Local Decomposition of the Canonical Height
Archimedean Absolute Values —— Explicit Formulas
Non-Archimedean Absolute Values —— Explicit Formulas
Exercises

APPENDIX A
Some Useful Tables
Bernoulli Numbers and (2k)
Fourier Coefficients of A(T) and j(T)
Elliptic Curves over Q with Complex Multiplication
Notes on Exercises
References
List of Notation
Index