Customer Reviews

Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics)

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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.

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......

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.

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. They have also found applications to more practical subjects, such as cryptography, so it's fair to say that national security and modern finance depend, at least in part, on what was at one point very abstract, theoretical mathematics. These are just a few reasons why an undergraduate studying mathematics might care to learn about elliptic curves.

The most modern treatments of ECs require a relatively sophisticated background. One not only needs to know basic algebra, topology, and complex analysis, but also commutative algebra, algebraic number theory, and a bit of algebraic geometry. The great advantage of this book is that it pares down the required background to almost the bare minimum. Readers are expected to know the content of an introductory algebra course (groups, rings, and fields for the final chapter), and some elementary number theory, but not much more. Certainly algebraic geometry is not required, and whatever projective geometry is needed is provided in a helpful appendix. There are also many exercises in the book, most of which are of an appropriate difficulty. A small percentage of the problems require substantial dedication and ingenuity to solve, and will serve to challenge most readers.

The principal drawback of the accessibility is that theorems are not proven in quite as much generality as one might like. For example, the Mordell-Weil theorem (the group of rational points on an EC is finitely generated), is really only proven for the special class of curves with a rational point of order 2, and Siegel's theorem on finiteness of integer points is only proven for a special class of cubic curves. Nevertheless, even these special cases have deep, serious proofs, and the more general proofs use the ideas behind the proofs in this book in a substantial way. The more accessible presentation of the material in this book will serve many as a good first step towards learning the general proofs of these theorems, should readers decide to continue learning more.

The only substantial gripe I have with this book are the errors in it. Most of these are minor, but some are quite substantial. Fortunately, Silverman maintains an errata at his website, which actually contains not just corrections but also features he may include in the next edition of this book, should he and Tate ever decide to update it. Overall, this book is a great place for undergraduates to start learning about a deep and important part of number theory, without needing lots of prerequisites. The book is also suitable for self-study, as it is well-written, not particularly long, and has lots of useful exercises.

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. Their conversational style really works.

This is a topic about which I knew almost nothing before-hand, apart from some basic stuff on elliptic curve cryptography (which is not covered here.) Now, at least, I start to see the tip of the iceburg on such a beautiful subject.