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椭圆曲线算术中的高等论题 英文版【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

椭圆曲线算术中的高等论题 英文版
  • JosephH.Silverman著 著
  • 出版社： 世界图书北京出版公司
  • ISBN：9787510004834
  • 出版时间：2010
  • 标注页数：528页
  • 文件大小：111MB
  • 文件页数：538页
  • 主题词：椭圆曲线－研究－英文

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图书目录

Introduction1

CHAPTER Ⅰ Elliptic and Modular Functions5

1．The Modular Group6

2．The Modular Curve X(1)14

3．Modular Functions23

4．Uniformization and Fields of Moduli34

5．Elliptic Functions Revisited38

6．q-Expansions of Elliptic Functions47

7．q-Expansions of Modular Functions55

8．Jacobi's Product Formula for △(τ)62

9．Hecke Operators67

10．Hecke Operators Acting on Modular Forms74

11．L-Series Attached to Modular Forms80

Exercises85

CHAPTER Ⅱ Complex Multiplication95

1．Complex Multiplication over C96

2．Rationality Questions104

3．Class Field Theory—A Brief Review115

4．The Hilbert Class Field121

5．The Maximal Abelian Extension128

6．Integrality of j140

7．Cyclotomic Class Field Theory151

8．The Main Theorem of Complex Multiplication157

9．The Associated Gr？ssencharacter165

10．The L-Series Attached to a CM Elliptic Curve171

Exercises178

CHAPTER Ⅲ Elliptic Surfaces187

1．Elliptic Curves over Function Fields188

2．The Weak Mordell-Weil Theorem191

3．Elliptic Surfaces200

4．Heights on Elliptic Curves over Function Fields212

5．Split Elliptic Surfaces and Sets of Bounded Height220

6．The Mordell-Weil Theorem for Function Fields230

7．The Geometry of Algebraic Surfaces231

8．The Geometry of Fibered Surfaces236

9．The Geometry of Elliptic Surfaces245

10．Heights and Divisors on Varieties255

11．Specialization Theorems for Elliptic Surfaces265

12．Integral Points on Elliptic Curves over Function Fields274

Exercises278

CHAPTER Ⅳ The Néron Model289

1．Group Varieties290

2．Schemes and S-Schemes297

3．Group Schemes306

4．Arithmetic Surfaces311

5．Néron Models318

6．Existence of Néron Models325

7．Intersection Theory,Minimal Models,and Blowing-Up338

8．The Special Fiber of a Néron Model350

9．Tate's Algorithm to Compute the Special Fiber361

10．The Conductor of an Elliptic Curve379

11．Ogg's Formula389

Exercises396

CHAPTER Ⅴ Elliptic Curves over Complete Fields408

1．Elliptic Curves over C408

2．Elliptic Curves over R413

3．The Tate Curve422

4．The Tate Map Is Surjective429

5．Elliptic Curves over p-adic Fields438

6．Some Applications of p-adic Uniformization445

Exercises448

CHAPTER Ⅵ Local Height Functions454

1．Existence of Local Height Functions455

2．Local Decomposition of the Canonical Height461

3．Archimedean Absolute Values—Explicit Formulas463

4．Non-Archimedean Absolute Values—Explicit Formulas469

Exercises476

APPENDIX A Some Useful Tables481

1．Bernoulli Numbers and ζ(2k)481

2．Fourier Coefficients of △(τ) and j(τ)482

3．Elliptic Curves over Q with Complex Multiplication483

Notes on Exercises484

References488

List of Notation498

Index504

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