Machine generated contents note: pt. I DEGREE
ch. 1 Degree of a Curve
1.Greek Mathematics
2.Degree
3.Parametric Equations
4.Our Two Definitions of Degree Clash
ch. 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
ch. 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
ch. 4 Multiplicities and Degree
1.Curves as Varieties
2.Multiplicities
3.Intersection Multiplicities
4.Calculus for Dummies
ch. 5 Bezout's Theorem
1.A Sketch of the Proof
2.An Illuminating Example
pt. II ELLIPTIC CURVES AND ALGEBRA
ch. 6 Transition to Elliptic Curves
ch. 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
ch. 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
5.Two Numerical Examples
ch. 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
ch. 10 Elliptic Curves over Q
1.The Basic Structure of the Group
2.Torsion Points
3.Points of Infinite Order
4.Examples
pt. III ELLIPTIC CURVES AND ANALYSIS
ch. 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
ch. 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
ch. 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
ch. 14 Surprising Properties of L-functions
1.Compare and Contrast
2.Analytic Continuation
3.Functional Equation
ch. 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?.