Theprefacetoatextbookfrequentlycontainstheauthor'sjustificationforofferingthepublic"anotherbook"onthegivensubject.Forourchosentopic,thearithmeticofellipticcurves,thereislittleneedforsuchanapologia.Consideringthevastamountofresearchcurrentlybeingdoneinthisarea,thepaucityofintroductorytextsissomewhatsurprising.PartsofthetheoryarecontainedinvariousbooksofLang(especially[La3]and[La5]);andtherearebooksofKoblitz([Ko])andRobert([Rob],nowoutofprint)whichconcentratemostlyontheanalyticandmodulartheory.Ina

Contents

Preface

Introduction

CHAPTER1

Algebraic Varieties

?. Affine Varieties

?. Projective Varieties

?. Maps between Varieties

CHAPTER11

Algebraic Curves

?. Curves

?. Maps between Curves

?. Divisors

?. DifTerentials

?. The Riemann-Roch Theorem

CHAPTER III

The Geometry of Elliptic Curves

?. Weierstrass Equations

?. The Group Law

?. Elliptic Curves

?. Isogenies

?. The Invariant Differential

?. The Dual Isogeny

?. The Tate Module

?. The Weil Pairing

?. The Endomorphism Ring

?0. The Automorphism Group

CHAPTERIV

The Formal Group of an Elliptic Curve

?. Expansion around 0

?. Formal Groups

?. Groups Associated to Formal Groups

?. The Invariant DifTerential

?. The Formal Logarithm

?. Fonnal Groups over Discrete Valuation Rings

?. Formal Groups in Characteristic p

CHAPTER V

Elliptic Curves over Finite Fields

?. Number of Rational Points

?. The Weil Conjectures

?. The Endomorphism Ring

?. Calculating the Hasse Invariant

CHAPTER VI

Elliptic Curves over C

?. Elliptic Integrals

?. Elliptic Functions

?. Construction of Elliptic Functions

?. Maps-Analytic and Algebraic

?. Uniformization

?. The Lefschetz Principle

CHAPTER VII

Elliptic Curves over Local Fields

?. Minimal Weierstrass Equations

?. Reduction Modulo

?. Points of Finite Order

?. The Action of Inertia

?. Good and Bad Reduction

?. The Group E/E0

?. The Criterion of Neron-Ogg-Shafarevich

CHAPTER VIII

Elliptic Curves over Global Fields

?. The Weak Mordell-Weil Theorem

?. The Kummer Pairing via Cohomology

?. The Descent Procedure

?. The Mordell-Weil Theorem over Q

?. Heights on Projective Space

?. Heights on Elliptic Curves

?. Torsion Points

?. The Minimal Discriminant

?. The Canonical Height

?0. TheRankofan Elliptic Curve

CHAPTERIX

Integral Points on Elliptic Curves

?. Diophantine Approximation

?. Distance Functions

?. Siegel's Theorem

?. The S-Unit Equation

?. EfTective Methods

?. Shafarevich's Theorem

?. The Curve Y2 = X3 + D

?. Roth's Theorem-An Overview

CHAPTER X

Computing the Mordell-Weil Group

?. An Example

?. Twisting--General Theory

?. Homogeneous Spaces

?. The Selmer and Shafarevich-Tate Groups

?. Twisting-Elliptic Curves

?. The Curve Y2 = X3 + DX

APPENDIX A

Elliptic Curves in Characteristics 2 and 3

APPENDIX B

Group Cohomology (H0 and H1)

?. Cohomology of Finite Groups

?. Galois Cohomology

?. Non-Abelian Cohomology

APPENDIX C

Further Topics: An Overview

?1. Complex Multiplication

?2. Modular Functions

?3. Modular Curves

?4. Tate Curves

?5. Neron Models and Tate's Algorithm

?6. L-Series

?7. Duality Theory

?8. Local Height Functions

?9. TheImageofGalois

?0. Function Fields and Specialization Theorems

Notes on Exercises

Bibliography

List ofNotation

Index