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Rational Points on Elliptic Curves
Joseph H. Silverman and John Tate
Publisher: Springer (1994)
Details: 281 pages, Hardcover
Series: Undergraduate Texts in Mathematics
Topics: Number Theory, Elliptic Curves, Arithmetic Algebraic Geometry, Algebraic Number Theory
This book is in the MAA's basic library list.
MAA Review[Reviewed by Allen Stenger, on 08/22/2008]
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 MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
BLL*** — The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.