The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.
Rational Points on Elliptic Curves
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Editorial Reviews
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From the reviews: "The authors' goal has been to write a textbook in a technically difficult field which is accessible to the average undergraduate mathematics major, and it seems that they have succeeded admirably..."--MATHEMATICAL REVIEWS "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. … The examples really pull together the material and make it clear. … a great book for a first introduction to the subject of elliptic curves. … very clearly written and you will understand a lot when you are done." (Allen Stenger, The Mathematical Association of America, August, 2008)
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29 of 29 people found the following review helpful
By A Customer
Format:Hardcover
This wonderful book is an excellent introduction to elliptic curves over the rational numbers. It is self-contained and easily accessible, but still takes the reader quite far, thus giving an undergraduate reader some exciting glimpes of deeper mathematics. This book is ideally suited as a text book for an undergraduate course (I have myself enjoyed it as a course), but is written in a lively style that also makes it fun to read on one's own. It covers such topics as the Nagell-Lutz Theorem, Mordell's Theorem over rational numbers, elliptic curves over finite fields and reduction modulo p, Thue's Theorem and diophantine approximation, and even an introduction to complex multiplication. An appendix provides the reader with a basic background on projective geometry. This book is a must for any student wanting to see beyond the ordinary coursework, and at the same time provides a natural stepping stone to a more advanced treatment of the subject, such as "The Arithmetic of Elliptic Curves", also by Silverman, which has become pretty much the standard text on the subject.
15 of 15 people found the following review helpful
Format:Hardcover
The authors do a fantastic job of introducing elliptic curves for individuals and students interested in this area. Because of the importance of elliptic curves to cryptography, in integrable models in statistical mechanics, in superstring theory in physics, in mirror symmetry in algebraic geometry, in mechanics in the solution of the spinning top, and even in financial engineering, this book will be useful in building intuition about these interesting objects. Be careful in reading this book though...the theory of elliptic curves is beautiful and addicting, and you will want no doubt to read more about them after finishing it. There are two other books by Silverman that will alleviate the monkey on your back for more knowledge about elliptic curves. Happy reading......
14 of 16 people found the following review helpful
Format:Hardcover
The virtue of this book is its leisurely style; and the subject is very attractive as well. The idea that lies behind this whole theory is the way in which one imposes an arithmetic on the points of an elliptic curve. There is much to be said about this: the whole thing becomes a group, the rational points form a finitely generated group, etc. That's chapters 1-3. In chapters 4-5 we finally get to some number-theoretical applications. First applications to factorisation (and thus to cryptography). Then chapter 5 is on "integer points on cubic curves", which sounds nice and classical, but things quickly turn quite messy. And finally there is chapter 6, on complex multiplication and Kronecker's Jugendtraum, for those who are interested in this extension field business that everybody seems to love. One could certainly read chapter 1-3 as a very elementary and easy-going introduction to the basics of elliptic curve arithmetic. I think the applications are treated better elsewhere, in Koblitz's books for instance.
Most Recent Customer Reviews
5.0 out of 5 stars
Good introduction to elliptic curves suitable for undergraduates
Elliptic curves have come to occupy a central place in number theory, as the spectacular proof of Fermat's Last Theorem fifteen years ago showed. Read more
Published on May 17, 2010 by Nona
5.0 out of 5 stars
"It is possible to write endlessly on elliptic curves. (This is not a...
This is great. I wish the prose in more "tougher" maths texts were written in similarly. The math can get so tough (for me) so quickly, but the authors are very gentle. Read more
Published on July 8, 2008 by M_A_S
