橢圓曲線理論是代數(shù)、幾何、分析和數(shù)論的混合體，書中在講述基本理論的同時(shí)強(qiáng)調(diào)各部分之間的相互作用，以便讀者更好的學(xué)習(xí)現(xiàn)代數(shù)學(xué)的精髓。本書的可讀性強(qiáng)，寫作風(fēng)格自由，配合大量的練習(xí)使得本書成為對(duì)Diophantine方程和算術(shù)幾何感興趣的讀者的理想選擇。
目次：幾何和算術(shù)；有限階點(diǎn)；有理點(diǎn)群；有限域上的三次曲線；三次曲線上的整數(shù)點(diǎn)；復(fù)數(shù)乘法；射影幾何。
讀者對(duì)象：數(shù)學(xué)專業(yè)的本科生、研究生和相關(guān)的讀者。
Preface
Computer Packages
Acknowlments
Introduction
CHAPTER 1
Geometry and Arithmetic
1.Rational Points on Conics
2.The Geometry of Cubic Curves
3.Weierstrass Normal Form
4.Explicit Formulas for the Group Law
Exercises
CHAPTER 2
Points of Finite Order
1.Points of Order Two and Three
2.Real and Complex Points on Cubic Curves
3.The Discriminant
4.Points of Finite Order Have Integer Coordinates
5.The Nagell—Lutz Theorem and Further Developments
Exercises
CHAPTER 3
The Group of Rational Points
1.Heights and Descent
2.The Height of P + P0
3.The Height of 2P
4.A Useful Homomorphism
5.Mordell's Tneorem
6.Examples and Further Developments
7.Singular Cubic Curves
Exercises
CHAPTER 4
Cubic Curves over Finite Fields
1.Rational Points over Finite Fields
2.A Theorem of Gauss
3.Points of Finite Order Revisited
4.A Factorization Algorithm Using Elliptic Curves
Exercises
CHAPTER 5
Integer Points on Cubic Curves
1.How Many Integer Points？
2.Taxicabs and Sums of Two Cubes
3.Thue's Theorem and Diophantine Approximation
4.Construction of an Auxiliary Polynomial
5.The Auxiliary Polynomialls Small
6.The Auxiliary Polynomial Does Not Vanish
7.Proof of the Diophantine Approximation Theorem
8.Further Developments
Exercises
CHAPTER 6
Complex Multiplication
1.Abelian Extensions of Q
2.Algebraic Points on Cubic Curves
3.A Galois Representation
4.Complex Multiplication
5.Abelian Extensions of Q（i）
Exercises
APPENDIX A
Projective Geometry
1.Homogeneous Coordinates and the Projective Plane
2.Curves in the Projective Plane
3.Intersections of Projective Curves
4.Intersection Multiplicities and a Proof of Bezout's Theorem
5.Reduction Modulo p
Exercises
Bibliography
List of Notation
Index
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