Elliptic curves, modular forms, and their L-functions /

The topic of elliptic curves and their applications in number theory has developed widely in the last 15 years since Andrew Wiles's success in proving Fermat's last theorem using related methods. Several good books on elliptic curves are in print, including those of Joseph Silverman and John Tate, Anthony Knapp, and J. S. Milne. The book under review offers a survey, based on lectures for undergraduates held at the Park City Institute. In so short a book, completeness is impossible. This survey is successful because Lozano-Robledo (Univ. of Connecticut) has a keen sense of what is most important to present, presents it well, and gives carefully chosen explicit examples and computations that endow the presentation with a real immediacy. The three broad topics of the book's title are united by the Taniyama-Shimura-Weil conjecture, proven by Wiles and his coworkers, and they feature in the Birch and Swinnerton-Dyer conjecture that is described at the end of the main part of the book. A welcome addition to a serious mathematics library. Summing Up: Recommended. Lower-division undergraduates through researchers/faculty. J. McCleary Vassar College