Rational Points on Elliptic CurvesIn 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. |
Contents
| 1 | |
| 9 | |
The Geometry of Cubic Curves | 15 |
Weierstrass Normal Form | 22 |
Explicit Formulas for the Group Law | 28 |
CHAPTER II | 38 |
The Discriminant | 47 |
The NagellLutz Theorem and Further Developments | 56 |
CHAPTER V | 145 |
Thues Theorem and Diophantine Approximation | 152 |
The Auxiliary Polynomial Is Small | 165 |
Proof of the Diophantine Approximation Theorem | 171 |
CHAPTER VI | 180 |
A Galois Representation | 193 |
Complex Multiplication | 199 |
Abelian Extensions of Qi | 205 |
CHAPTER III | 63 |
The Height of 2P | 71 |
Mordells Theorem | 83 |
Examples and Further Developments | 89 |
Singular Cubic Curves | 99 |
Cubic Curves over Finite Fields | 107 |
Points of Finite Order Revisited | 121 |
APPENDIX | 220 |
Intersections of Projective Curves | 233 |
Intersection Multiplicities and a Proof of Bezouts Theorem | 242 |
Reduction Modulo p | 251 |
Bibliography | 259 |
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Common terms and phrases
abelian group algebraic algorithm Auxiliary Polynomial ax² b₁ Bezout's theorem C₁ C1 and C2 C₂ Chapter complex numbers compute conic cubic curve cubic equation curve y² cyclic group defined denominator Diophantine duplication formula element elliptic curve equation y² example Fermat's field finite number finite order Galois extension Galois group geometry gives group law group of order group of rational height homogeneous homomorphism integer coefficients integer coordinates integer solutions isomorphism Lemma look mod Q*2 modulo Mordell's theorem Nagell-Lutz theorem non-singular number theory one-to-one order three P₁ point at infinity points of finite points of order polynomial f(x projective plane prove quadratic rational numbers rational points rational solutions real numbers roots satisfy Section solutions in integers subgroup Suppose tangent line Weierstrass Weierstrass equation zero