The Arithmetic of Elliptic Curves

Joseph H. Silverman (auth.)0387962034, 9780387962030

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, elliptic curves over finite fields, the complex numbers, local fields, and global fields. The last two chapters deal with integral and rational points, including Siegel’s theorem and explicit computations for the curve Y^2 = X^3 + DX.
The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

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Table of contents :
Front Matter….Pages i-xii
Introduction….Pages 1-4
Algebraic Varieties….Pages 5-20
Algebraic Curves….Pages 21-44
The Geometry of Elliptic Curves….Pages 45-109
The Formal Group of an Elliptic Curve….Pages 110-129
Elliptic Curves over Finite Fields….Pages 130-145
Elliptic Curves over ℂ….Pages 146-170
Elliptic Curves over Local Fields….Pages 171-188
Elliptic Curves over Global Fields….Pages 189-240
Integral Points on Elliptic Curves….Pages 241-275
Computing the Mordell-Weil Group….Pages 276-323
Back Matter….Pages 324-402