Publisher's Synopsis
The emphasis of this text is on the number-theoretic aspects of elliptic curves. Using an informal style, the authors attempt to present a mathematically difficult field in a readable manner.;The first part is devoted to proving the fundamental theorems of the field (or at least special cases of these): The Nagell-Lutz theorem, Mordell's theorem, and Hasse's theorem.;The remainder of the book discusses special topics and newer developments. A discussion of Lara's algorithm for factoring large numbers shows an application of elliptic curves to the "real world", in this case, the problem of public-key cryptographic systems. A proof of Siegel's theorem, which asserts that an elliptic curve has only a finite number of integer points, serves to introduce the powerful notions of Diophantine approximation techniques. A final chapter introduces the theory of complex multiplication and discusses how points of finite order on elliptic curves can be used to generate extension fields with Abelian Galois groups.;The book can readily be used for a one-semester course; parts of it can also serve as the basis for a supplementary topic at the end of a traditional course in either aalgebraic geometry or number theory. Many exercises are included, ranging from easy calculations to the published theorems.
