Graduate Texts in Mathematics, 84. New York etc.: Springer-Verlag. xiv, 389 p. DM 98.00 (1990).

This is a somewhat expanded version of the first edition that appeared in 1982 [Springer Graduate texts in mathematics 84, (1982;

Zbl 0482.10001). The first edition of 1982 consisted of 18 chapters which have been included in the second edition without alterations. The second edition contains two new chapters, one on the Mordell-Weil theorem for elliptic curves and one on recent developments in arithmetic geometry. The authors wrote the first edition with the purpose to give insight into modern developments in number theory by showing their close relationship with classical, 19th century number theory. The authors wrote the new chapters 19 and 20 in the same spirit. In chapter 19, they give a proof of the Mordell-Weil theorem for elliptic curves over ${\bbfQ}$ without using Kummer theory or algebraic geometry: they first give Cassels’ proof of the weak Mordell-Weil theorem which uses only a weaker version of Dirichlet’s unit theorem for number fields; and then they derive the Mordell-Weil theorem using the standard descent argument which is worked out by elementary arithmetic. Chapter 19 is meant as a preparation for chapter 20, in which the authors give a very interesting overview of the important developments in arithmetic geometry after the appearance of the first edition of their book. Among other things, they discuss the Mordell conjecture proved by Faltings, the Taniyama-Weil conjecture and the result of Frey, Serre and Ribet that this implies Fermat’s last theorem, recent progress on the Birch-Swinnerton-Dyer conjecture by Coates-Wiles, Gross-Zagier, Rubin, and Kolyvagin, and the derivation of Gauss’ class number conjecture from the results of Gross-Zagier. In chapter 20, the authors do not give proofs but they give sufficient background to understand and appreciate the results. Chapter 20 is an excellent introduction for those who want to study the subject more thoroughly.