Preface to the Second Edition
Preface to the First Edition
Acknowledgments to the Second Edition
Acknowledgments to the First Edition
Introduction to Rational Points on Plane Curves
1 Rational Lines in the Projective Plane
2 Rational Points on Conics
3 Pythagoras, Diophantus, and Fermat
4 Rational Cubics and Mordeli's Theorem
5 The Group Law on Cubic Curves and Elliptic Curves
6 Rational Points on Rational Curves. Faltings and the Mordell Conjecture
7 Real and Complex Points on EUipticCurves
8 The Elliptic Cu~e. Group Law on the Intersection of Two Quadrics in
Projective.Three Space
1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve
2 Plane Algebraic Curves
3 Elliptic Curves and Their Isomorphisms
4 Families of Elliptic Curves and Geometric Propertiesof Torsion Points
5 Reduction mod p and Torsion Points
6 Proof of Mordell's Finite Generation Theorem.
7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields
8 Descent and Galois Cohomology
9 Elliptic and Hypergeometric Functions
10 Theta Functions
11 Modular Functions.
12 Endomorphisms of Elliptic Curves
13 Elliptic Curves over Finite Fields
14 Elliptic Curves over Local Fields
15 Elliptic Curves over Global Fields and e-Adic Representations
16 L.Function of an,Elliptic Curve and Its Analytic Continuation
17 Remarks on the Birch and Swinnerton-Dyer Conjecture
18 Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem
19 Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties
20 Families of Elliptic Curves
Appendix
References
List of Notation
Index
Divides naturally into several parts according to the level of the material,the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level,the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level.
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