The theory of elliptic curves & modular forms provides a fruitful meeting ground for such diverse areas as number theory complex analysis algebraic geometry & representation theory This book starts out with a problem from elementary number theory & proceeds to lead its reader into the modern theory covering such topics as the Hasse-Weil L-function & the conjecture of Birch & Swinnerton-Dyer This new edition details the current state of knowledge of elliptic curves