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Introduction to Rational Points on Plane Curves 1

1 Rational Lines in the Projective Plane 2

2 Rational Points on Conics 4

3 Pythagoras,Diophantus,and Fermat 7

4 Rational Cubics and Mordell's Theorem 10

5 The Group Law on Cubic Curves and Elliptic Curves 13

6 Rational Points on Rational Curves.Faltings and the Mordell Conjecture 17

7 Real and Complex Points on Elliptic Curves 19

8 The Elliptic Curve Group Law on the Intersection of Two Quadrics in Projective Three Space 20

1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve 23

1 Chord-Tangent Computational Methods on a Normal Cubic Curve 23

2 Illustrations of the Elliptic Curve Group Law 28

3 The Curves with Equations y2=x3十ax and y2=x3+a 34

4 Multiplication by 2 on an Elliptic Curve 38

5 Remarks on the Group Law on Singular Cubics 41

2 Plane Algebraic Curves 45

1 Projective Spaces 45

2 Irreducible Plane Algebraic Curves and Hypersurfaces 47

3 Elements of Intersection Theory for Plane Curves 50

4 Multiple or Singular Points 52

Appendix to Chapter 2:Factorial Rings and Elimination Theory 57

1 Divisibility Properties of Factorial Rings 57

2 Factorial Properties of Polynomial Rings 59

3 Remarks on Valuations and Algebraic Curves 60

4 Resultant of Two Polynomials 61

3 Elliptic Curves and Their Isomorphisms 65

1 The Group Law on a Nonsingular Cubic 65

2 Normal Forms for Cubic Curves 67

3 The Discriminant and the Invariantj 70

4 Isomorphism Classification in Characteristics≠2,3 73

5 Isomorphism Classification in Characteristic 3 75

6 Isomorphism Classification in Characteristic 2 76

7 Singular Cubic Curves 80

8 Parameterization of Curves in Characteristic Unequal to 2 or 3 82

4 Families of Elliptic Curves and Geometric Properties of Torsion Points 85

1 The Legendre Family 85

2 Families of Curves with Points of Order 3:The Hessian Family 88

3 The Jacobi Family 91

4 Tate's Normal Form for a Cubic with a Torsion Point 92

5 An Explicit 2-Isogeny 95

6 Examples of Noncyclic Subgroups of Torsion Points 101

5 Reduction mod P and Torsion Points 103

1 Reduction mod P of Projective Space and Curves 103

2 Minimal Normal Forms for an Elliptic Curve 106

3 Good Reduction of Elliptic Curves 109

4 The Kernel of Reduction mod P and the P-Adic Filtration 111

5 Torsion in Elliptic Curves over Q:Nagell-Lutz Theorem 115

6 Computability of Torsion Points on Elliptic Curves from Integrality and Divisibility Properties of Coordinates 118

7 Bad Reduction and Potentially Good Reduction 120

8 Tate's Theorem on Good Reduction over the Rational Numbers 122

6 Proof of Mordell's Finite Generation Theorem 125

1 A Condition for Finite Generation of an Abelian Group 125

2 Fermat Descent and x4+y4=1 127

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 Maps 132

6 The General Notion of Height on Projective Space 135

7 The Canonical Height and Norm on an Elliptic Curve 137

8 The Canonical Height on Projective Spaces over Global Fields 140

7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields 143

1 Galois Theory:Theorems of Dedekind and Artin 143

2 Group Actions on Sets and Groups 146

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

4 Long Exact Sequence in G-Cohomology 151

5 Some Calculations with Galois Cohomology 153

6 Galois Cohomology Classification of Curves with Given j-Invariant 155

8 Descent and Galois Cohomology 157

1 Homogeneous Spaces over Elliptic Curves 157

2 Primitive Descent Formalism 160

3 Basic Descent Formalism 163

9 Elliptic and Hypergeometric Functions 167

1 Quotients of the Complex Plane by Discrete Subgroups 167

2 Generalities on Elliptic Functions 169

3 The Weierstrass ？-Function 171

4 The Differential Equation for ？（z） 174

5 Preliminaries on Hypergeometric Functions 179

6 Periods Associated with Elliptic Curves:Elliptic Integrals 183

10 Theta Functions 189

1 Jacobi q-Parametrization:Application to Real Curves 189

2 Introduction to Theta Functions 193

3 Embeddings of a Torus by Theta Functions 195

4 Relation Between Theta Functions and Elliptic Functions 197

5 The Tate Curve 198

6 Introduction to Tate's Theory of P-Adic Theta Functions 203

11 Modular Functions 209

1 Isomorphism and Isogeny Classification of Complex Tori 209

2 Families of Elliptic Curves with Additional Structures 211

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

4 Modular Functions 220

5 The L-Function of a Modular Form 222

6 Elementary Properties of Euler Products 224

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

8 Hecke Operators:New Forms 229

9 Modular Polynomials and the Modular Equation 230

12 Endomorphisms of Elliptic Curves 233

1 Isogenies and Division Points for Complex Tori 233

2 Symplectic Pairings on Lattices and Division Points 235

3 Isogenies in the General Case 237

4 Endomorphisms and Complex Multiplication 241

5 The Tate Module of an Elliptic Curve 245

6 Endomorphisms and the Tate Module 246

7 Expansions Near the Origin and the Formal Group 248

13 Elliptic Curves over Finite Fields 253

1 The Riemann Hypothesis for Elliptic Curves over a Finite Field 253

2 Generalities on Zeta Functions of Curves over a Finite Field 256

3 Definition of Supersingular Elliptic Curves 259

4 Number of Supersingular Elliptic Curves 263

5 Points ofOrder P and Supersingular Curves 265

6 The Endomorphism Algebra and Supersingular Curves 266

7 Summary of Criteriafor a Curve To Be Supersingular 268

8 Tate's Description of Homomorphisms 270

9 Division Polynomial 272

14 Elliptic Curves over Local Fields 275

1 The Canonical P-Adic Filtration on the Points of an Elliptic Curve over a Local Field 275

2 The Néron Minimal Model 277

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

4 Elliptic Curves over the Real Numbers 284

15 Elliptic Curves over Global Fields and e-Adic Representations 291

1 Minimal Discriminant Normal Cubic Forms over a Dedekind Ring 291

2 Generalities on ？-Adic Representations 293

3 Galois Representations and the Néron-Ogg-Safarevi？Criterion in the Global Case 296

4 Ramification Properties of ？-Adic ReDresentations of Number Fields:Cebotarev's Density Theorem 298

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

6 Weight Properties of Frobenius Elements in ？-Adic Representations:Faltings'Finiteness Theorem 303

7 Tate's Conjecture,Safarevi？'s Theorem,and Faltings'Proof 305

8 Image of ？-Adic Representations of Elliptic Curves:Serre's Open Image Theorem 307

16 L-Function of an Elliptic Curve and Its Analytic Continuation 309

1 Remarks on Analytic Methods in Arithmetic 309

2 Zeta Functions ofCurves over Q 310

3 Hasse-Weil L-Function and the Functional Equation 312

4 Classical Abelian L-Functions and Their Functional Equations 315

5 Gr？ssencharacters and Hecke L-Functions 318

6 Deuring's Theorem on the L-Function of an Elliptic Curve with Complex Multiplication 321

7 Eichler-Shimura Theory 322

8 The Modular Curve Conjecture 324

17 Remarks on the Birch and Swinnerton-Dyer Conjecture 325

1 The Conjecture Relating Rank and Order of Zero 325

2 Rank Conjecture for Curves with Complex Multiplication Ⅰ,by Coates and Wiles 326

3 Rank Conjecture for Curves with Complex Multiplication Ⅱ,byGreenberg and Rohrlich 327

4 Rank Coniecture for Modular Curves bv Gross and Zagier 328

5 Goldfeld's Work on the Class Number Problem and Its Relation to the Birch and Swinnerton-Dyer Conjecture 328

6 The Conjecture of Birch and Swinnerton-Dyer on the Leading Term 329

7 Heegner Points and the Derivative of the L-function at s=1,after Gross and Zagier 330

8 Remarks On Postscript:October 1986 331

18 Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem 333

1 Semistable Curves and Tate Modules 334

2 The Frey Curve and the Reduction of Fermat Equation to Modular Elliptic Curves over Q 335

3 Modular Elliptic Cuives and the Hecke Algebra 336

4 Hecke Algebras and Tate Modules of Modular Elliptic Curves 338

5 Special Properties of mod 3 Representations 339

6 Deformation Theory and ？-Adic Representations 339

7 Properties of the Universal Deformation Ring 341

8 Remarks on the Proof of the Opposite Inequality 342

9 Survey of the Nonsemistable Case of the Modular Curve Conjecture 342

19 Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties 345

1 Smooth Manifolds:Real Differential Geometry 347

2 Complex Analytic Manifolds:Complex Differential Geometry 349

3 K？hler Manifolds 352

4 Connections,Curvature,and Holonomy 356

5 Proiective Spaces,Characteristic Classes,and Curvature 361

6 Characterizations of Calabi-Yau Manifolds:First Examples 366

7 Examples of Calabi-Yau Varieties from Toric Geometry 369

8 Line Bundles and Divisors:Picard and Néron-Severi Groups 371

9 Numerical Invariants of Surfaces 374

10 Enriques Classification for Surfaces 377

11 Introduction to K3 Surfaces 378

20 Families of Elliptic Curves 383

1 Algebraic and Analytic Geometry 384

2 Morphisms Into Projective Spaces Determined by Line Bundles,Divisors,and Linear Systems 387

3 Fibrations Especially Surfaces Over Curves 390

4 Generalities on Elliptic Fibrations of Surfaces Over Curves 392

5 Elliptic K3 Surfaces 395

6 Fibrations of 3 Dimensional Calabi-Yau Varieties 397

7 Three Examples of Three Dimensional Calabi-Yau Hypersurfaces in Weight Projective Four Space and Their Fibfings 400

Appendix Ⅰ:Calabi-Yau Manifolds and String Theory 403

Stefan Theisen 403

Why String Theory? 403

Basic Properties 404

String Theories in Ten Dimensions 406

Compactification 407

Duality 409

Summary 411

Appendix Ⅱ:Elliptic Curves in Algorithmic Number Theory and Cryptography 413

Otto Forster 413

1 Applications in Algorithmic Number Theory 413

1.1 Factorization 413

1.2 Deterministic Primality Tests 415

2 Elliptic Curves in Cryptography 417

2.1 The Discrete Logarithm 417

2.2 Diffie-Hellman Key Exchange 417

2.3 Digital Signatures 418

2.4 Algorithms for the Discrete Logarithm 419

2.5 Counting the Number of Points 421

2.6 Schoof's Algorithm 421

2.7 Elkies Primes 423

References 424

Appendix Ⅲ:Elliptic Curves and Topological Modular Forms 425

1 Categories in a Category 427

2 Groupoids in a Category 429

3 Cocategories over Commutative Algebras:Hopf Algebroids 431

4 The Category WT（R）and the Weierstrass Hopf Algebroid 434

5 Morphisms of Hopf Algebroids:Modular Forms 438

6 The Role of the Formal Group in the Relation Between Elliptic Curves and General Cohomology Theory 441

7 The Cohomology Theory or Spectrum tmf 443

References 444

Appendix Ⅳ:Guide to the Exercises 445

Ruth Lawrence 465

References 465

List of Notation 479

Index 481