This is a corrected and revised edition of the author's wonderful 1992 book \ref[first edition, 1992; MR 93m:11053]. As in the first edition, the author provides a clear and thorough description of the major algorithms needed to investigate elliptic curves defined over the rational numbers. The book is in three parts. The first part is devoted to the computation of modular symbols, modular forms, modular parametrization, and special values of $L$-series. It includes worked examples for curves of conductor $N=11$, $33$, $37$, and $49$. Part two gives basic algorithms for studying the arithmetic of elliptic curves over $\bold Q$, including minimal models, torsion points, descent and Mordell-Weil groups, canonical heights, periods, isogenies, twists, and complex multiplication. The final part describes all (modular) elliptic curves of conductor $N\le999$, including equations for the curves, generators for their Mordell-Weil groups, eigenvalues of Hecke operators, data related to the conjecture of Birch and Swinnerton-Dyer, and modular parametrization degrees (the latter a new feature in this edition). All of this extends considerably the original Antwerp IV list of curves of conductor $N\le200$ \ref[ Modular functions of one variable. IV, Lecture Notes in Math., 476, Springer, Berlin, 1975; MR 51 #12708]. This new edition describes many significant improvements in the algorithms (too many to detail here). Further, the author has rewritten all of the algorithms in C{\tt++} and made them available in his {\tt mwrank} package, available on the web at www.maths.ex.ac.uk/\char"7E cremona/packages.html. He has also extended his calculations to all (modular) elliptic curves of conductor up to $N=5077$, but since the data is so voluminous, it is not included in the book, but is available on the above web site.

Reviewed by Joseph H. Silverman

REVIEW OF FIRST EDITION

Let $\bold Q$ be the field of rational numbers and $N$ a positive integer. In this book the author describes in detail the algorithm, based on modular symbols, for computing all the modular elliptic curves defined over $\bold Q$ with conductor $N$, and for each of these curves explains the method of computing the quantities entering in the Birch and Swinnerton-Dyer conjectures (torsion part, rank and generators of the Mordell-Weil group, regulator, traces of Frobenius, leading coefficient of the $L$-series at $s=1$, Kodaira symbols and local factors). Although the idea of using modular symbols for computing the modular elliptic curves defined over $\bold Q$ with conductor $N$ is not new, neither the complete description of the algorithm nor the description of its implementation had been available before the writing of this book; moreover, the complete list of all the modular curves defined over $\bold Q$ with conductor less than 999, and the computation of all these curves' interesting arithmetic quantities (i.e. the continuation of the "Antwerp IV" famous tables), which are given at the end of the book, will prove very useful for any mathematician interested in the arithmetic of elliptic curves. The book is divided into three sections. In the first section the modular symbol method is described. It is assumed that the reader knows the theory of modular symbols; the author's main point is to present in detail every step that has to be performed in order to find explicitly all the newforms of conductor $N$, the sign of the functional equation of their $L$-functions, the (rational) quotients of the value of these functions at $1$ divided by certain periods, the value at $1$ of the first derivative of these $L$-functions which is not zero at $1$, and the eigenvalues of the Hecke operators acting on these newforms. In a short appendix, the cases $N=11,33,37,49$ are worked out as an illustration of the method. In the second part the author starts with a Weierstrass model of an elliptic curve defined over $\bold Q$; he describes in detail the computations that have to be performed for obtaining the arithmetic quantities attached to this curve that enter in the Birch and Swinnerton-Dyer conjectures, and their implementations; this means the computation of the torsion part of the Mordell-Weil group (using Mazur's bounds for torsion over $\bold Q$), generators of the free part of the Mordell-Weil group (using $2$-descent, which turns out to work for all the curves whose conductor is less than 999), the regulator (using the local decomposition of the canonical height), the computation of all isogenous curves over $\bold Q$ (using Mazur's bound for isogeny over $\bold Q$), and the way of recognising twists. The third section is the presentation of the actual result of the work, that is, the extension of the "Antwerp IV" tables for conductors up to 999 (without any attempt to verify that no curves of conductor less than 999 are missing in the table, i.e., that there are no nonmodular elliptic curves defined over $\bold Q$ whose conductor is less than 999). Of course the table is presented so that it can be used without any knowledge of the first two parts.

Reviewed by Philippe Satgé