Cover

Title

Copyright

Contents

Preface

Acknowledgments

Prologue

PART I: DEGREE

Chapter 1 Degree of a Curve

1. Greek Mathematics

2. Degree

3. Parametric Equations

4. Our Two Definitions of Degree Clash

Chapter 2 Algebraic Closures

1. Square Roots of Minus One

2. Complex Arithmetic

3. Rings and Fields

4. Complex Numbers and Solving Equations

5. Congruences

6. Arithmetic Modulo a Prime

7. Algebraic Closure

Chapter 3 The Projective Plane

1. Points at Infinity

2. Projective Coordinates on a Line

3. Projective Coordinates on a Plane

4. Algebraic Curves and Points at Infinity

5. Homogenization of Projective Curves

6. Coordinate Patches

Chapter 4 Multiplicities and Degree

1. Curves as Varieties

2. Multiplicities

3. Intersection Multiplicities

4. Calculus for Dummies

Chapter 5 Bézout’s Theorem

1. A Sketch of the Proof

2. An Illuminating Example

PART II: ELLIPTIC CURVES AND ALGEBRA

Chapter 6 Transition to Elliptic Curves

Chapter 7 Abelian Groups

1. How Big Is Infinity?

2. What Is an Abelian Group?

3. Generations

4. Torsion

5. Pulling Rank

Appendix: An Interesting Example of Rank and Torsion

Chapter 8 Nonsingular Cubic Equations

1. The Group Law

2. Transformations

3. The Discriminant

4. Algebraic Details of the Group Law

5. Numerical Examples

6. Topology

7. Other Important Facts about Elliptic Curves

8. Two Numerical Examples

Chapter 9 Singular Cubics

1. The Singular Point and the Group Law

2. The Coordinates of the Singular Point

3. Additive Reduction

4. Split Multiplicative Reduction

5. Nonsplit Multiplicative Reduction

6. Counting Points

7. Conclusion

Appendix A: Changing the Coordinates of the Singular Point

Appendix B: Additive Reduction in Detail

Appendix C: Split Multiplicative Reduction in Detail

Appendix D: Nonsplit Multiplicative Reduction in Detail

Chapter 10 Elliptic Curves over Q

1. The Basic Structure of the Group

2. Torsion Points

3. Points of Infinite Order

4. Examples

PART III: ELLIPTIC CURVES AND ANALYSIS

Chapter 11 Building Functions

1. Generating Functions

2. Dirichlet Series

3. The Riemann Zeta-Function

4. Functional Equations

5. Euler Products

6. Build Your Own Zeta-Function

Chapter 12 Analytic Continuation

1. A Difference that Makes a Difference

2. Taylor Made

3. Analytic Functions

4. Analytic Continuation

5. Zeroes, Poles, and the Leading Coefficient

Chapter 13 L-functions

1. A Fertile Idea

2. The Hasse-Weil Zeta-Function

3. The L-Function of a Curve

4. The L-Function of an Elliptic Curve

5. Other L-Functions

Chapter 14 Surprising Properties of L-functions

1. Compare and Contrast

2. Analytic Continuation

3. Functional Equation

Chapter 15 The Conjecture of Birch and Swinnerton-Dyer

1. How Big Is Big?

2. Influences of the Rank on the Np’s

3. How Small Is Zero?

4. The BSD Conjecture

5. Computational Evidence for BSD

6. The Congruent Number Problem

Epilogue

Retrospect

Where Do We Go from Here?

Bibliography

Index

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