93j:11032
Knapp, Anthony W.
Elliptic curves.
Mathematical Notes, 40.
Princeton University Press, Princeton, NJ, 1992. xvi+427 pp. \$24.95. ISBN 0-691-08559-5
11G05 (11-01 11F11 14H52)

References: 0 Reference Citations: 13 Review Citations: 1
In recent years there has been widespread interest in the arithmetic theory of elliptic curves and the related theory of modular functions and curves. A number of introductory books in the subject have been published in the past decade, including some by N. Koblitz [ Introduction to elliptic curves and modular forms, Springer, New York, 1984; MR 86c:11040], the reviewer \ref[ The arithmetic of elliptic curves, Springer, New York, 1986; MR 87g:11070], D. Husemoller [ Elliptic curves, Springer, New York, 1987; MR 88h:11039], and the reviewer and J. T. Tate \ref[ Rational points on elliptic curves, Springer, New York, 1992; MR 93g:11003]. The book under review has some features in common with each of these books, but has many distinctive features of its own which make it a welcome addition to the literature.

The author's book can be divided into two fairly distinct parts. The first part deals with the arithmetic theory of elliptic curves defined over $ Q$, including proofs of Mordell's theorem $(E( Q)$ is finitely generated) and the Lutz-Nagell theorem (points of finite order have integer coordinates with $y^2$ dividing the discriminant). The proofs follow the standard path, but by restricting attention to $ Q$ the author avoids a number of technicalities which are required in the general case. This will be appreciated by the reader learning the subject for the first time. The author also intersperses the text with many instructive examples and remarks.

The second part of the book investigates the theory of modular functions, modular forms, and modular curves. It is somewhat surprising, in view of the vast amount of work on the arithmetic theory of modular curves done in the past 20 years, that there is no elementary (i.e., mid-level graduate) text which presents the basic material. The author's book goes a long way towards filling that gap. Although he is not able to include full proofs of all of the requisite background material, such as the theory of abelian varieties and Jacobians, he always gives a clear explanation of the results he needs, and full references are provided.

In summary, the first half of this book is an excellent place for beginners to learn the basics about the arithmetic theory of elliptic curves over $ Q$, especially if they lack the background in algebraic number theory needed for a more advanced text. And the second half of the book, providing as it does the only available down-to-earth introduction to the theory of modular curves, is sure to be a standard source for anyone who wants to study this fascinating subject. Our only criticism is that the author has not included exercises, since most students find it easiest to learn a subject by working on problems. On the positive side, the author has included very informative notes for each chapter which will be invaluable for the reader wishing to further pursue some topic.

The following list of chapter titles provides a further description of the contents of this book: I. Overview; II. Curves in projective space; III. Cubic curves in Weierstrass form; IV. Mordell's theorem; V. Torsion subgroup of $E( Q)$; VI. Complex points; VII. Dirichlet's theorem; VIII. Modular forms over ${\rm SL}(2, Z)$; IX. Modular forms for Hecke subgroups; X. $L$ function of an elliptic curve; XI.\ Eichler-Shimura theory; XII. Taniyama-Weil conjecture.

Reviewed by Joseph H. Silverman