Graduate Texts in Mathematics 111. New York, NY: Springer (ISBN 0-387-95490-2/hbk). xxi, 487 p. EUR 84.95/net; sFr. 144.00; £ 65.50; $ 79.95 (2004).

The first edition of this meanwhile standard text on elliptic curves was published in 1987 (

Zbl 0605.14032). Intended as an all-round introduction to the geometry and arithmetic of elliptic curves, the text divided naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. The first edition came with seventeen chapters that led the reader from the most elementary aspects of cubic equations, with full proofs given, up to the, by then, most advanced topics in the theory of elliptic curves, where the presentation became rather survey-like and sketchy. Nevertheless, Husemöller’s text was and is a great first introduction to the world of elliptic curves, in their various aspects and trends, and a good guide to the current research literature as well.
The book under review is the second, up-dated and considerably enlarged edition of this standard text. Increased by about forty percent in volume, mainly by the addition of three new chapters and a few new sections to the old ones, this second edition builds on the original in several ways. Moreover, there are now three appendices which are partly written by co-authors, and which point out further recent developments and applications within the theory of elliptic curves.
The three new chapters are chapters 18, 19, and 20. The author provides here surveys of recent directions and extensions of the theory, thereby focusing on concepts, statements of new results, explanations of their strategy of proof, and relations to the more classic parts of the theory of elliptic curves.
In this vein, chapter 18 is devoted to the recent achievements towards Fermat’s last theorem. As the author points out, this chapter is primarily designed to point out which material in the earlier chapters is relevant as background for studying {\it A. Wiles}’s original work on the subject [Modular elliptic curves and Fermat’s last theorem, Ann. Math. (2) 141, 443--551 (1995;

Zbl 0823.11029)], together with the further developments due to the work of F. Diamond and R. Taylor in the late 1990s.
Basically, chapter 18 provides an introduction to the various research papers on the so-called modular curve conjecture (or Shimura-Taniyama-Weil conjecture) in survey form, with many comments and hints, but without proofs.
Chapter 19 deals with Calabi-Yau varieties as higher-dimensional analogues of elliptic curves. After a quick compilation of the relevant topics from Hermitian and Kählerian differential geometry, the various characterizations of Calabi-Yau varieties are explained and illustrated by examples. At the end of this survey, the Enriques classification for surfaces and a brief introduction to $K3$-surfaces is sketched. As the author points out, this chapter has been added for the sake of utility of the book for people interested in examples of fibrations of three-dimensional Calabi-Yau varieties. In view of the very fact that arithmetical Calabi-Yau varieties have recently become important in physics, especially in conformal field theory, the addition of this chapter may be seen as a service to a wider public, too.
Chapter 20 returns to the earlier material on families of elliptic curves, this time in the context of modern algebraic geometry (i.e., the theory of schemes) and analytic geometry. The author’s main goal is here to point out some of the many areas of mathematics (and physics) in which families of elliptic curves play an important role. This includes, apart from a summary of the relevant concepts in algebraic and analytic theory, another survey on surfaces over curves, elliptic $K3$-surfaces, fibrations of 3-dimensional Calabi-Yau varieties, and examples of 3-dimensional Calabi-Yau hypersurfaces in weighted projective 4-space and their fiberings. In this regard, the new chapter 20 enhances the earlier material on elliptic fibrations by additional, more topical examples.
As to the new three appendices, the first one has been contributed by {\it S. Theisen}. Entitled “Calabi-Yau Manifolds and String Theory”, this short essay gives a physicist’s view to Calabi-Yau varieties from a rather philosophical standpoint, thereby enfilading the mathematical discussion from this lookout.
Appendix II was written by {\it O. Forster} and depicts “Elliptic Curves in Algorithmic Number Theory and Cryptography”. This survey adds a discussion of the use of elliptic curves in arithmetical computing theory and coding theory to the overall panorama that the author of the book was striving for.
Appendix III, written by {\it D. Husemöller} himself, provides another panoramic outlook, namely an elementary introduction to the relation between elliptic curves (given by Weierstrass cubics) and the homological (or homotopical) theory of modular forms. The central concept is here the so-called Weierstrass Hopf algebroid, in the categorical sense, and it is illustrated how this framework can be used to compute the homotopy of the so-called spectrum topological modular forms. The development of this topic is still in progress, and its discussion here leads the reader directly to the forefront of current research in the field.
Each appendix comes with its own list of special bibliographical references and according hints to them.
Apart from these enlargements of the book under review, the author has also taken the opportunity to make numerous minor additions to the original text, which has otherwise been left totally intact. Now as before, there are numerous exercises in each of the more elementary chapters, and there is again that (now fourth) appendix by {\it R. Lawrence} providing solutions for them.
Of course, the bibliography has been up-dated and enlarged, too, and that by about 75 new references.
All together, this second edition of D. Husemöller’s standard text on elliptic curves offers a much broader panoramic view to the subject than the first one, without having lost its introductory character with respect to the elementary parts, and it has certainly gained a good deal of topicality, appeal, power of inspiration, and educational value for a wider public. No doubt, this text will maintain its role as both a useful primer and a passionate invitation to the evergreen theory of elliptic curves and their applications.