Elliptic Curves. (MN-40)

Rating:
Summary: So much insight for so little assumed background
Review: Usually the Princeton Notes are written for specialists in the field. This book by Knapp is a definite exception. The didactic quality is excellent, and the background assumed of the reader is meager compared to what you can take away by reading this book. The reader is only assumed to have an undergraduate background in complex analysis and modern algebra...not bad considering the frontiers that Knapp reaches in this book. The Eichler-Shimura thoery is treated in the next to last chapter of the book, and at a very understandable level. After reading the book one takes away an understanding of many deep results in the theory of elliptic curves. The author gives the reader a rare gift for most modern mathematical texts: insight. Proofs are given for most of the results, but the main emphasis is on understanding how elliptic curves are studied and why they are useful mathematical objects. This is definitely a book to be read by everyone interested in the theory of elliptic curves. String theorists, cryptoanalysts, physicists, and aspiring mathematicians whose interests and applications are in the theory of elliptic curves should definitely read this book. A mere $50.00 will get you a copy...but it is definitely worth four times that. If you are coming to the theory of elliptic curves for the first time and have a background in comnplex analysis and modern algebra, read first Joseph Silverman's book on rational points on elliptic curves, and then Knapp's book. Then move on to Silverman's two books on the arithmetic of elliptic curves. For both theory and applications, this sequence should prepare anyone for entering this very interesting world of elliptic curves. If only more books in modern mathematics were written like Knapp's Elliptic Curves...............