Rational points on elliptic curves

Table of Contents

1.Geometry and Arithmetic
1.1.Rational Points on Conies
1.2.The Geometry of Cubic Curves
1.3.Weierstrass Normal Form
1.4.Explicit Formulas for the Group Law
Exercises
2.Points of Finite Order
2.1.Points of Order Two and Three
2.2.Real and Complex Points on Cubic Curves
2.3.The Discriminant
2.4.Points of Finite Order Have Integer Coordinates
2.5.The Nagell
Lutz Theorem and Further Developments
3.The Group of Rational Points
3.1.Heights and Descent
3.2.The Height of P + P0
3.3.The Height of 2P
3.4.A Useful Homomorphism
3.5.Mordell's Theorem
3.6.Examples and Further Developments
3.7.Singular Cubic Curves
Exercises Ill
4.Cubic Curves over Finite Fields
4.1.Rational Points over Finite Fields
4.2.A Theorem of Gauss
4.3.Points of Finite Order Revisited
4.4.A Factorization Algorithm Using Elliptic Curves
4.5.Elliptic Curve Cryptography
Contents note continued: 5.Integer Points on Cubic Curves
5.1.How Many Integer Points?
5.2.Taxicabs and Sums of Two Cubes
5.3.Thue's Theorem and Diophantine Approximation
5.4.Construction of an Auxiliary Polynomial
5.5.The Auxiliary Polynomial Is Small
5.6.The Auxiliary Polynomial Does Not Vanish
5.7.Proof of the Diophantine Approximation Theorem
5.8.Further Developments
6.Complex Multiplication
6.1.Abelian Extensions of Q
6.2.Algebraic Points on Cubic Curves
6.3.A Galois Representation
6.4.Complex Multiplication
6.5.Abelian Extensions of Q(i)
6.6.Elliptic Curves and Fermat's Last Theorem
A.Projective Geometry
A.1.Homogeneous Coordinates and the Projective Plane
A.2.Curves in the Projective Plane
A.3.Intersections of Projective Curves
A.4.Intersection Multiplicities and a Proof of Bezout's Theorem
A.5.Reduction Modulo p
B.Transformation to Weierstrass Form.

In 1961 the second author deliv1lred a series of lectures at Haverford Col­ lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran­ scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por­ tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter­ est in the theory of elliptic curves for subjects ranging from cryptogra­ phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig­ inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove.

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