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Rational Points on Elliptic Curves

Joseph H. Silverman and John Tate
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Allen Stenger
, on

This is a very leisurely introduction to the theory of elliptic curves, concentrating on an algebraic and number-theoretic viewpoint. It is pitched at an undergraduate level and simplifies the work by proving the main theorems with additional hypotheses or by only proving special cases.

This is an introductory text, but after the first chapter it is a disparate collection of topics (some more disparate than others). There's a clear path through the first three chapters, which focus on determining the structure of the group of rational points. The main result here is the Nagell-Lutz theorem that all points of finite order have integer coordinates, and the y-coordinate always divides the discriminant of the cubic. This makes it easy to determine all the points of finite order for any given equation. These chapters also prove Mordell's theorem that the group of rational points is a finitely-generated abelian group. Chapter 3 concludes with an excellent section of examples of determining the group structure for several particular elliptic curves. The examples really pull together the material and make it clear.

Roughly the middle third of the book is aimed at Siegel's theorem that a non-singular cubic curve with integer coefficients has only finitely many points with integer coordinates. The text a special case of Thue's equation, namely ax3 + by3 = c. There is also an excursion into elliptic curve factorization methods.

The last part of the book didn't fit as well as the rest; it deals with complex multiplication of algebraic points on cubic curves. The book concludes with a lengthy appendix on projective and algebraic geometry (which also did not fit in well).

This is a great book for a first introduction to the subject of elliptic curves. It doesn't cover as much ground as Silverman's Arithmetic of Elliptic Curves or Koblitz's Introduction to Elliptic Curves and Modular Forms, but it is very clearly written and you will understand a lot when you are done.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.



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CHAPTER I — Geometry and Arithmetic
1. Rational Points on Conics
2. The Geometry of Cubic Curves
3. Weierstrass Normal Form
4. Explicit Formulas for the Group Law

CHAPTER II — Points of Finite Order
1. Points of Order Two and Three
2. Real and Complex Points on Cubic Curves
3. The Discriminant
4. Points of Finite Order Have Integer Coordinates
5. The Nagell-Lutz Theorem and Further Developments

CHAPTER III — The Group of Rational Points
1. Heights and Descent
2. The Height of P + P_0
3. The Height of 2P
4. A Useful Homomorphism
5. Mordell's Theorem
6. Examples and Further Developments
7. Singular Cubic Curves

CHAPTER IV — Cubic Curves over Finite Fields
1. Rational Points over Finite Fields
2. A Theorem of Gauss
3. Points of Finite Order Revisited
4. A Factorization Algorithm Using Elliptic Curves

CHAPTER V — Integer Points on Cubic Curves
1. How Many Integer Points?
2. Taxicabs and Sums of Two Cubes
3. Thue's Theorem and Diophantine Approximation
4. Construction of an Auxiliary Polynomial
5. The Auxiliary Polynomial Is Small
6. The Auxiliary Polynomial Does Not Vanish
7. Proof of the Diophantine Approximation Theorem
8. Further Developments

CHAPTER VI — Complex Multiplication
1. Abelian Extensions of Q
2. Algebraic Points on Cubic Curves
3. A Galois Representation
4. Complex Multiplication
5. Abelian Extensions of Q(i)

APPENDIX A — Projective Geometry
1. Homogeneous Coordinates and the Projective Plane
2. Curves in the Projective Plane
3. Intersections of Projective Curves
4. Intersection Multiplicities and a Proof of Bezout's Theorem
5. Reduction Modulo p


List of Notation