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Introduction to Rational Points on Plane Curves1

1 Rational Lines in the Projective Plane2

2 Rational Points on Conics4

3 Pythagoras,Diophantus,and Fermat7

4 Rational Cubics and Mordell's Theorem10

5 The Group Law on Cubic Curves and Elliptic Curves13

6 Rational Points on Rational Curves.Faltings and the Mordell Conjecture17

7 Real and Complex Points on Elliptic Curves19

8 The Elliptic Curve Group Law on the Intersection of Two Quadrics in Projective Three Space20

1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve23

1 Chord-Tangent Computational Methods on a Normal Cubic Curve23

2 Illustrations of the Elliptic Curve Group Law28

3 The Curves with Equations y2=x3十ax and y2=x3+a34

4 Multiplication by 2 on an Elliptic Curve38

5 Remarks on the Group Law on Singular Cubics41

2 Plane Algebraic Curves45

1 Projective Spaces45

2 Irreducible Plane Algebraic Curves and Hypersurfaces47

3 Elements of Intersection Theory for Plane Curves50

4 Multiple or Singular Points52

Appendix to Chapter 2:Factorial Rings and Elimination Theory57

1 Divisibility Properties of Factorial Rings57

2 Factorial Properties of Polynomial Rings59

3 Remarks on Valuations and Algebraic Curves60

4 Resultant of Two Polynomials61

3 Elliptic Curves and Their Isomorphisms65

1 The Group Law on a Nonsingular Cubic65

2 Normal Forms for Cubic Curves67

3 The Discriminant and the Invariantj70

4 Isomorphism Classification in Characteristics≠2,373

5 Isomorphism Classification in Characteristic 375

6 Isomorphism Classification in Characteristic 276

7 Singular Cubic Curves80

8 Parameterization of Curves in Characteristic Unequal to 2 or 382

4 Families of Elliptic Curves and Geometric Properties of Torsion Points85

1 The Legendre Family85

2 Families of Curves with Points of Order 3:The Hessian Family88

3 The Jacobi Family91

4 Tate's Normal Form for a Cubic with a Torsion Point92

5 An Explicit 2-Isogeny95

6 Examples of Noncyclic Subgroups of Torsion Points101

5 Reduction mod P and Torsion Points103

1 Reduction mod P of Projective Space and Curves103

2 Minimal Normal Forms for an Elliptic Curve106

3 Good Reduction of Elliptic Curves109

4 The Kernel of Reduction mod P and the P-Adic Filtration111

5 Torsion in Elliptic Curves over Q:Nagell-Lutz Theorem115

6 Computability of Torsion Points on Elliptic Curves from Integrality and Divisibility Properties of Coordinates118

7 Bad Reduction and Potentially Good Reduction120

8 Tate's Theorem on Good Reduction over the Rational Numbers122

6 Proof of Mordell's Finite Generation Theorem125

1 A Condition for Finite Generation of an Abelian Group125

2 Fermat Descent and x4+y4=1127

3 Finiteness of（E（Q）:2E（Q））for E=E[a,b]128

4 Finiteness of the Index（E（k）:2E（k））129

5 Quasilinear and Quasiquadratic Maps132

6 The General Notion of Height on Projective Space135

7 The Canonical Height and Norm on an Elliptic Curve137

8 The Canonical Height on Projective Spaces over Global Fields140

7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields143

1 Galois Theory:Theorems of Dedekind and Artin143

2 Group Actions on Sets and Groups146

3 Principal Homogeneous G-Sets and the First Cohomology Set H1（G,A）148

4 Long Exact Sequence in G-Cohomology151

5 Some Calculations with Galois Cohomology153

6 Galois Cohomology Classification of Curves with Given j-Invariant155

8 Descent and Galois Cohomology157

1 Homogeneous Spaces over Elliptic Curves157

2 Primitive Descent Formalism160

3 Basic Descent Formalism163

9 Elliptic and Hypergeometric Functions167

1 Quotients of the Complex Plane by Discrete Subgroups167

2 Generalities on Elliptic Functions169

3 The Weierstrass ？-Function171

4 The Differential Equation for ？（z）174

5 Preliminaries on Hypergeometric Functions179

6 Periods Associated with Elliptic Curves:Elliptic Integrals183

10 Theta Functions189

1 Jacobi q-Parametrization:Application to Real Curves189

2 Introduction to Theta Functions193

3 Embeddings of a Torus by Theta Functions195

4 Relation Between Theta Functions and Elliptic Functions197

5 The Tate Curve198

6 Introduction to Tate's Theory of P-Adic Theta Functions203

11 Modular Functions209

1 Isomorphism and Isogeny Classification of Complex Tori209

2 Families of Elliptic Curves with Additional Structures211

3 The ModularCurvesX（N）,X1（N）,and X0（N）215

4 Modular Functions220

5 The L-Function of a Modular Form222

6 Elementary Properties of Euler Products224

7 Modular Forms for Г0（N）,Г1（N）,and г（N）227

8 Hecke Operators:New Forms229

9 Modular Polynomials and the Modular Equation230

12 Endomorphisms of Elliptic Curves233

1 Isogenies and Division Points for Complex Tori233

2 Symplectic Pairings on Lattices and Division Points235

3 Isogenies in the General Case237

4 Endomorphisms and Complex Multiplication241

5 The Tate Module of an Elliptic Curve245

6 Endomorphisms and the Tate Module246

7 Expansions Near the Origin and the Formal Group248

13 Elliptic Curves over Finite Fields253

1 The Riemann Hypothesis for Elliptic Curves over a Finite Field253

2 Generalities on Zeta Functions of Curves over a Finite Field256

3 Definition of Supersingular Elliptic Curves259

4 Number of Supersingular Elliptic Curves263

5 Points ofOrder P and Supersingular Curves265

6 The Endomorphism Algebra and Supersingular Curves266

7 Summary of Criteriafor a Curve To Be Supersingular268

8 Tate's Description of Homomorphisms270

9 Division Polynomial272

14 Elliptic Curves over Local Fields275

1 The Canonical P-Adic Filtration on the Points of an Elliptic Curve over a Local Field275

2 The Néron Minimal Model277

3 Galois Criterion of Good Reduction of Néron-Ogg-Safarevi？280

4 Elliptic Curves over the Real Numbers284

15 Elliptic Curves over Global Fields and e-Adic Representations291

1 Minimal Discriminant Normal Cubic Forms over a Dedekind Ring291

2 Generalities on ？-Adic Representations293

3 Galois Representations and the Néron-Ogg-Safarevi？Criterion in the Global Case296

4 Ramification Properties of ？-Adic ReDresentations of Number Fields:Cebotarev's Density Theorem298

5 Rationality Properties of Frobenius Elements in ？-Adic Representations:Variation of ？301

6 Weight Properties of Frobenius Elements in ？-Adic Representations:Faltings'Finiteness Theorem303

7 Tate's Conjecture,Safarevi？'s Theorem,and Faltings'Proof305

8 Image of ？-Adic Representations of Elliptic Curves:Serre's Open Image Theorem307

16 L-Function of an Elliptic Curve and Its Analytic Continuation309

1 Remarks on Analytic Methods in Arithmetic309

2 Zeta Functions ofCurves over Q310

3 Hasse-Weil L-Function and the Functional Equation312

4 Classical Abelian L-Functions and Their Functional Equations315

5 Gr？ssencharacters and Hecke L-Functions318

6 Deuring's Theorem on the L-Function of an Elliptic Curve with Complex Multiplication321

7 Eichler-Shimura Theory322

8 The Modular Curve Conjecture324

17 Remarks on the Birch and Swinnerton-Dyer Conjecture325

1 The Conjecture Relating Rank and Order of Zero325

2 Rank Conjecture for Curves with Complex Multiplication Ⅰ,by Coates and Wiles326

3 Rank Conjecture for Curves with Complex Multiplication Ⅱ,byGreenberg and Rohrlich327

4 Rank Coniecture for Modular Curves bv Gross and Zagier328

5 Goldfeld's Work on the Class Number Problem and Its Relation to the Birch and Swinnerton-Dyer Conjecture328

6 The Conjecture of Birch and Swinnerton-Dyer on the Leading Term329

7 Heegner Points and the Derivative of the L-function at s=1,after Gross and Zagier330

8 Remarks On Postscript:October 1986331

18 Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem333

1 Semistable Curves and Tate Modules334

2 The Frey Curve and the Reduction of Fermat Equation to Modular Elliptic Curves over Q335

3 Modular Elliptic Cuives and the Hecke Algebra336

4 Hecke Algebras and Tate Modules of Modular Elliptic Curves338

5 Special Properties of mod 3 Representations339

6 Deformation Theory and ？-Adic Representations339

7 Properties of the Universal Deformation Ring341

8 Remarks on the Proof of the Opposite Inequality342

9 Survey of the Nonsemistable Case of the Modular Curve Conjecture342

19 Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties345

1 Smooth Manifolds:Real Differential Geometry347

2 Complex Analytic Manifolds:Complex Differential Geometry349

3 K？hler Manifolds352

4 Connections,Curvature,and Holonomy356

5 Proiective Spaces,Characteristic Classes,and Curvature361

6 Characterizations of Calabi-Yau Manifolds:First Examples366

7 Examples of Calabi-Yau Varieties from Toric Geometry369

8 Line Bundles and Divisors:Picard and Néron-Severi Groups371

9 Numerical Invariants of Surfaces374

10 Enriques Classification for Surfaces377

11 Introduction to K3 Surfaces378

20 Families of Elliptic Curves383

1 Algebraic and Analytic Geometry384

2 Morphisms Into Projective Spaces Determined by Line Bundles,Divisors,and Linear Systems387

3 Fibrations Especially Surfaces Over Curves390

4 Generalities on Elliptic Fibrations of Surfaces Over Curves392

5 Elliptic K3 Surfaces395

6 Fibrations of 3 Dimensional Calabi-Yau Varieties397

7 Three Examples of Three Dimensional Calabi-Yau Hypersurfaces in Weight Projective Four Space and Their Fibfings400

Appendix Ⅰ:Calabi-Yau Manifolds and String Theory403

Stefan Theisen403

Why String Theory?403

Basic Properties404

String Theories in Ten Dimensions406

Compactification407

Duality409

Summary411

Appendix Ⅱ:Elliptic Curves in Algorithmic Number Theory and Cryptography413

Otto Forster413

1 Applications in Algorithmic Number Theory413

1.1 Factorization413

1.2 Deterministic Primality Tests415

2 Elliptic Curves in Cryptography417

2.1 The Discrete Logarithm417

2.2 Diffie-Hellman Key Exchange417

2.3 Digital Signatures418

2.4 Algorithms for the Discrete Logarithm419

2.5 Counting the Number of Points421

2.6 Schoof's Algorithm421

2.7 Elkies Primes423

References424

Appendix Ⅲ:Elliptic Curves and Topological Modular Forms425

1 Categories in a Category427

2 Groupoids in a Category429

3 Cocategories over Commutative Algebras:Hopf Algebroids431

4 The Category WT（R）and the Weierstrass Hopf Algebroid434

5 Morphisms of Hopf Algebroids:Modular Forms438

6 The Role of the Formal Group in the Relation Between Elliptic Curves and General Cohomology Theory441

7 The Cohomology Theory or Spectrum tmf443

References444

Appendix Ⅳ:Guide to the Exercises445

Ruth Lawrence465

References465

List of Notation479

Index481