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  • 椭圆曲线算术中的高等论题(英文版)[平装]
  • 共2个商家     48.00元~54.60
  • 作者:西尔弗曼(JosephH.Silverman)(作者)
  • 出版社:世界图书出版公司;第1版(2010年1月1日)
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  • ISBN:9787510004834

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    《椭圆曲线算术中的高等论题(英文版)》是由世界图书出版公司出版的。

    目录

    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
    Jacobi's 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
    Tate's Algorithm to Compute the Special Fiber
    The Conductor of an Elliptic Curve
    Ogg's 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

    序言

    In the introduction to the first volume of The Arithmetic o/Elliptic Curves(Springer-Verlag, 1986), I observed that "the theory of elliptic curves isrich, varied, and amazingly vast," and as a consequence, "many importanttopics had to be omitted." I included a brief introduction to ten additionaltopics as an appendix to the first volume, with the tacit understanding thateventually there might be a second volume containing the details. You arenow holding that second volume.
    Unfortunately, it turned out that even those ten topics would not fitinto a single book, so I was forced to make some choices. The followingmaterial is covered in this book:
    I. Elliptic and modular functions for the full modular group.
    II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems.
    IV. Neron models, Kodaira-Neron classification of special fibers,
    Tate's algorithm, and Ogg's conductor-discriminant formula.
    V. Tate's theory of qcurves over p-adic fields.
    VI. Neron's theory of canonical local height functions.
    So what's still missing? First and foremost is the theory of modularcurves of higher level and the associated modular parametrizations of ellip-tic curves. There is little question that this is currently the hottest topicin the theory of elliptic curves, but any adequate treatment would seem torequire (at least) an entire book of its own. (For a nice introduction, seeKnapp [1].) Other topics that I have left out in order to keep this bookat a manageable size include the description of the image of the g-adicrepresentation attached to an elliptic curve and local and global dualitytheory. Thus, at best, this book covers approximately half of the materialdescribed in the appendix to the first volume. I apologize to those who mayfeel disappointed, either at the incompleteness or at the choice of particulartopics.

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