《椭圆曲线(第2版)》(作者胡斯迈勒)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.

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

i 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