Written by leading experts in the field
Presents a technically difficult field which is readable by the average undergraduate mathematics major
Brings the classic first edition up-to-date with the inclusion of hot topics such as elliptic curve cryptography and the ABC Conjecture
Explains the role of Frey curves and Galois representations in Andrew Wiles' breakthrough proof of Fermat's Last Theorem
Includes many new exercises

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry.

Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.

Topics
Algebraic Geometry
Number Theory

Data Structures, Cryptology and Information Theory

由该领域的主要专家撰写

介绍了一个技术上很难的领域，但普通本科数学专业的学生也能读懂。

将经典的第一版更新，加入了椭圆曲线密码学和ABC猜想等热门话题

解释了弗雷曲线和伽罗瓦表示在安德鲁-怀尔斯对费马最后定理的突破性证明中的作用

包括许多新的练习

椭圆曲线的理论涉及代数、几何、分析和数论的愉快融合。本卷在发展基本理论时强调了这种相互作用，从而为高年级本科生提供了一个欣赏现代数学的统一性的机会。同时，我们尽力只使用本科生课程中常用的方法和结果。这种可及性、非正式的写作风格和丰富的练习使《椭圆曲线上的有理数点》成为对学习索非亚方程和算术几何感兴趣的各年级学生的理想入门书。

最具体地说，椭圆曲线是两个变量的三次方多项式的零点集合。如果该多项式具有有理系数，那么人们可以要求对那些坐标为整数或有理数的零点进行描述。这个数论问题正是《椭圆曲线上的有理点》的主要课题。涵盖的主题包括椭圆曲线的几何和群结构、描述有限阶的点的Nagell-Lutz定理、关于有理点群的有限生成的Mordell-Weil定理、关于整数点集合的有限性的Thue-Siegel定理、关于有限域中有坐标的点的计数定理、Lenstra的椭圆曲线分解算法，以及对复数乘法和与扭转点相关的伽罗瓦表示的讨论。第二版新增的主题包括对椭圆曲线密码学的介绍，以及对怀尔斯等人通过使用椭圆曲线对费马最后定理的惊人证明的简要讨论。

主题

代数几何学

数论

数据结构、密码学和信息论