Rational Points on Elliptic Curves

by Joseph H. Silverman, John T. Tate

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves . Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.

Genres: Mathematics, Number Theory, Nonfiction, Textbooks, Reference

332 pages, Hardcover

First published June 24, 1992

Reviews

[Dave Neary · Rating 4 out of 5]

Some dense math! I want to go back and review the exercises in chapters 3-6.

[Molidere · Rating 4 out of 5]

Solid undergrad level treatment of one of the more fashionable areas of mathematics. The majority of the book is comprehensible to someone entering their Second Year of University study (in the UK at least). Should even be approachable to sophisticated high-school students who have familiarised themselves with some Group Theory and the like. A little brisk at points but one is unlikely to find a better treatment at this level.

[Rebekah Mercer · Rating 5 out of 5]

It was fun to be reminded of proofs of infinite descent, and proofs involving localisation (looking at an object wrt each prime, then piercing together what you learned at the primes to understand the object in a more general sense). The walkthrough of parts of the proof of Fermat's Last Theorem was a fun conclusion, and the explanation of projective coordinates was really the best i've seen :)

[Owen Levens · Rating 5 out of 5]

Came. Hard.