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 -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- V -- W -- Z.