Preface | p. xiii |
Acknowledgments | p. xix |
Prologue | p. 1 |
Degree | |
Degree of a Curve | p. 13 |
Greek Mathematics | p. 13 |
Degree | p. 14 |
Parametric Equations | p. 20 |
Our Two Definitions of Degree Clash | p. 23 |
Algebraic Closures | p. 26 |
Square Roots of Minus One | p. 26 |
Complex Arithmetic | p. 28 |
Rings and Fields | p. 30 |
Complex Numbers and Solving Equations | p. 32 |
Congruences | p. 34 |
Arithmetic Modulo a Prime | p. 38 |
Algebraic Closure | p. 38 |
The Projective Plane | p. 42 |
Points at Infinity | p. 42 |
Projective Coordinates on a Line | p. 46 |
Projective Coordinates on a Plane | p. 50 |
Algebraic Curves and Points at Infinity | p. 54 |
Homogenization of Projective Curves | p. 56 |
Coordinate Patches | p. 61 |
Multiplicities and Degree | p. 67 |
Curves as Varieties | p. 67 |
Multiplicities | p. 69 |
Intersection Multiplicities | p. 72 |
Calculus for Dummies | p. 76 |
BézoutÆs Theorem | p. 82 |
A Sketch of the Proof | p. 82 |
An Illuminating Example | p. 88 |
Elliptic Curves and Algebra | |
Transition to Elliptic Curves | p. 95 |
Abelian Groups | p. 100 |
How Big Is Infinity? | p. 100 |
What Is an Abelian Group? | p. 101 |
Generations, | p. 103 |
Torsion | p. 106 |
Pulling Rank | p. 108 |
Appendix: An Interesting Example of Rank and Torsion | p. 110 |
Nonsingular Cubic Equations | p. 116 |
The Group Law | p. 116 |
Transformations | p. 119 |
The Discriminant | p. 121 |
Algebraic Details of the Group Law | p. 122 |
Numerical Examples | p. 125 |
Topology | p. 127 |
Other Important Facts about Elliptic Curves | p. 131 |
Two Numerical Examples | p. 133 |
Singular Cubics | p. 135 |
The Singular Point and the Group Law | p. 135 |
The Coordinates of the Singular Point | p. 136 |
Additive Reduction | p. 137 |
Split Multiplicative Reduction | p. 139 |
Nonsplit Multiplicative Reduction | p. 141 |
Counting Points | p. 145 |
Conclusion | p. 146 |
Changing the Coordinates of the Singular Point | p. 146 |
Additive Reduction in Detail | p. 147 |
Split Multiplicative Reduction in Detail | p. 149 |
Nonsplit Multiplicative Reduction in Detail | p. 150 |
Elliptic Curves over Q | p. 152 |
The Basic Structure of the Group | p. 152 |
Torsion Points | p. 153 |
Points of Infinite Order | p. 155 |
Examples | p. 156 |
Elliptic Curves and Analysis | |
Building Functions | p. 161 |
Generating Functions | p. 161 |
Dirichlet Series | p. 167 |
The Riemann Zeta-Function | p. 169 |
Functional Equations | p. 171 |
Euler Products | p. 174 |
Build Your Own Zeta-Function | p. 176 |
Analytic Continuation | p. 181 |
A Difference that Makes a Difference | p. 181 |
Taylor Made | p. 185 |
Analytic Functions | p. 187 |
Analytic Continuation | p. 192 |
Zeroes, Poles, and the Leading Coefficient | p. 196 |
L-Functions | p. 199 |
A Fertile Idea | p. 199 |
The Hasse-Weil Zeta-Function | p. 200 |
The L-Function of a Curve | p. 205 |
The L-Function of an Elliptic Curve | p. 207 |
Other L-Functions | p. 212 |
Surprising Properties of L-Functions | p. 215 |
Compare and Contrast | p. 215 |
Analytic Continuation | p. 220 |
Functional Equation | p. 221 |
The Conjecture of Birch and Swinnerton-Dyer | p. 225 |
How Big Is Big? | p. 225 |
Influences of the Rank on the NpÆs | p. 228 |
How Small Is Zero? | p. 232 |
The BSD Conjecture | p. 236 |
Computational Evidence for BSD | p. 238 |
The Congruent Number Problem | p. 240 |
Epilogue | p. 245 |
Retrospect | p. 245 |
Where Do We Go from Here? | p. 247 |
Bibliography | p. 249 |
Index | p. 251 |
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