In the preface to the first edition of this book I remarked on the paucity of intro-ductory texts devoted to the arithmetic of elliptic curves. That unfortunate state of affairs has long since been remedied with the publication of many volumes, among which may be mentioned books by Cassels.
目录
Preface to the Second Edition
Preface to the First Edition
Introduction
CHAPTER 1
Algebraic Varieties
1. Affine Varieties
2. Projective Varieties
3. Maps Between Varieties
Exercises
CHAPTER 2
Algebraic Curves
1. Curves
2. Maps Between Curves
3. Divisors
4. Differentials
5. The Riemarm-Roch Theorem
Exercises
CHAPTER 3
The Geometry of Elliptic Curves
1. Weierstrass Equations
2. The Group Law
3. Elliptic Curves
4. Isogenies
5. The Invariant Differential
6. The Dual Isogeny
7. The Tate Module
8. The Weil Pairing
9. The Endomorphism Ring
10. The Automorphism Group
Exercises
CHAPTER 4
The Formal Group of an Elliptic Curve
1. Expansion Around O
2. Formal Groups
3. Groups Associated to Formal Groups
4. The Invariant Differential
5. The Formal Logarithm
6. Formal Groups over Discrete Valuation Rings
7. Formal Groups in Characteristic p
Exercises
CHAPTER 5
Elliptic Curves over Finite Fields
1. Number of Rational Points
2. The Weil Conjectures
3. The Endomorphism Ring
4. Calculating the Hasse Invariant
Exercises
CHAPTER 6
Elliptic Curves over C
1. Elliptic Integrals
2. Elliptic Functions
3. Construction of Elliptic Functions
4. Maps Analytic and Maps Algebraic
5. Uniformization
6. The Lefschetz Principle
Exercises
CHAPTER 7
Computing the Mordell-Weil Group
CHAPTER 8
Algorithmic Aspects of Elliptic Curves
APPENDIX
