Elliptic Curves Book

	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
1	Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve	23
2	Plane Algebraic Curves	45
App. to Ch. 2	Factorial Rings and Elimination Theory	57
3	Elliptic Curves and Their Isomorphisms	65
4	Families of Elliptic Curves and Geometric Properties of Torsion Points	85
5	Reduction mod p and Torsion Points	103
6	Proof of Mordell's Finite Generation Theorem	125
7	Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields	143
8	Descent and Galois Cohomology	157
9	Elliptic and Hypergeometric Functions	167
10	Theta Functions	189
11	Modular Functions	209
12	Endomorphisms of Elliptic Curves	233
13	Elliptic Curves over Finite Fields	253
14	Elliptic Curves over Local Fields	275
15	Elliptic Curves over Global Fields and l-Adic Representations	291
16	L-Function of an Elliptic Curve and Its Analytic Continuation	309
17	Remarks on the Birch and Swinnerton-Dyer Conjecture	325
18	Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem	333
19	Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties	345
20	Families of Elliptic Curves	383
App. I	Calabi-Yau Manifolds and String Theory	403
App. II	Elliptic Curves in Algorithmic Number Theory and Cryptography	413
App. III	Elliptic Curves and Topological Modular Forms	425
App. IV	Guide to the Exercises	445
	References	465
	List of Notation	479
	Index	481