椭圆曲线的算术理论（英文版）

The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic, the arithmetic of elliptic curves, there is little need for such an apologia.Considering the vast amount of research currently being done in this area,the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang (especially [La 3] and [La 5]); and there are books of Koblitz ([Ko]) and Robert ([Rob], now out of print) which concentrate mostly on the analytic and modular theory. In addition, survey articles have been written by Cassels ([Ca 7], really a short book) and Tate ([Ta 5]. which is beautifully written, but includes no proofs). Thus the author hopes that this volume will fill a real need, both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts.

The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic, the arithmetic of elliptic curves, there is little need for such an apologia.Considering the vast amount of research currently being done in this area,the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang (especially [La 3] and [La 5]); and there are books of Koblitz ([Ko]) and Robert ([Rob], now out of print) which concentrate mostly on the analytic and modular theory. In addition, survey articles have been written by Cassels ([Ca 7], really a short book) and Tate ([Ta 5]. which is beautifully written, but includes no proofs). Thus the author hopes that this volume will fill a real need, both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts.

　　本书为英文版。

Preface
Introduction
CHAPTER I 　Algebraic Varieties
　1. Affine Varieties
　2. Projective Varieties
　3. Maps between Varieties
CHAPTER II 　Algebraic Curves
　1. Curves
　2. Maps between Curves
　3. Divisors
　4. Differentials
　5. The Riemann-Roch Theorem
CHAPTER III 　The Geometry of Elliptic Curves
　1. Weierstrass Equations
　2. The Group Law
　3. Elliptic Curves
　4. Isogenies
　5. The Invariant Differential
　……
CHAPTER IV The Formal Group of an Elliptic Curve
CHAPTER V Elliptic Curves over Finite Fields
CHAPTER VI Elliptic Curves over
CHAPTER VII Elliptic Curves over Local Fields
CHAPTER VIII Elliptic Curves over Global Fields
CHAPTER IX Integral Points on Elliptic Curves
CHAPTER X Coputing the Mordell-Weil Group
APPENDIX
Note on Exercises
Bibliography
List of Notation
Index