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xvi Contents

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 Invariant j ........................... 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
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2 Fermat Descent and x + y = 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