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Tables of Contents for Rational Points on Elliptic Curves
Chapter/Section Title
Page #
Page Count
Preface
v
Computer Packages
vii
Acknowledgments
vii
Introduction
1
8
Geometry and Arithmetic
9
29
Rational Points on Conics
9
6
The Geometry of Cubic Curves
15
7
Weierstrass Normal Form
22
6
Explicit Formulas for the Group Law
28
10
Exercises
32
6
Points of Finite Order
38
25
Points of Order Two and Three
38
3
Real and Complex Points on Cubic Curves
41
6
The Discriminant
47
2
Points of Finite Order Have Integer Coordinates
49
7
The Nagell-Lutz Theorem and Further Developments
56
7
Exercises
58
5
The Group of Rational Points
63
44
Heights and Descent
63
5
The Height of P + P0
68
3
The Height of 2P
71
5
A Useful Homomorphism
76
7
Mordell's Theorem
83
6
Examples and Further Developments
89
10
Singular Cubic Curves
99
8
Exercises
102
5
Cubic Curves over Finite Fields
107
38
Rational Points over Finite Fields
107
3
A Theorem of Gauss
110
11
Points of Finite Order Revisited
121
4
A Factorization Algorithm Using Elliptic Curves
125
20
Exercises
138
7
Integer Points on Cubic Curves
145
35
How Many Integer Points?
145
2
Taxicabs and Sums of Two Cubes
147
5
Thue's Theorem and Diophantine Approximation
152
5
Construction of an Auxiliary Polynomial
157
8
The Auxiliary Polynomial Is Small
165
3
The Auxiliary Polynomial Does Not Vanish
168
3
Proof of the Diophantine Approximation Theorem
171
3
Further Developments
174
6
Exercises
177
3
Complex Multiplication
180
40
Abelian Extensions of Q
180
5
Algebraic Points on Cubic Curves
185
8
A Galois Representation
193
6
Complex Multiplication
199
6
Abelian Extensions of Q(i)
205
15
Exercises
213
7
APPENDIX A Projective Geometry
220
39
1. Homogeneous Coordinates and the Projective Plane
220
5
2. Curves in the Projective Plane
225
8
3. Intersections of Projective Curves
233
9
4. Intersections Multiplicities and a Proof of Bezout's Theorem
242
9
5. Reduction Modulo P
251
8
Exercises
254
5
Bibliography
259
4
List of Notation
263
4
Index
267