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Rational Points on Elliptic Curves (Undergraduate Texts in Mathematics) Corrected Edition
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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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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 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.