The classical theory of Diophantine equations concerns itself with the solution of systems of polynomial equations in rational numbers or integers. The phrase Diophantine geometry, coined by the author for the title of his highly influential 1962 monograph \ref[ Diophantine geometry, Wiley-Interscience, New York, 1962; MR 26 #119], refers to the study of Diophantine equations using methods of geometry. Originally, this meant using tools mainly from algebraic geometry, but now it includes (at least) the use of deep results from differential geometry and complex analysis. In the beautiful and wide-ranging book under review, the author attempts to survey this vast and rapidly changing subject. The goal is nothing less than to describe "the grand unification of algebraic geometry, analysis and PDE, Diophantine approximation, Nevanlinna theory, and classical Diophantine problems about rational and integral points". Given the enormity of such a task, the author succeeds to a remarkable extent. (The above quote, taken from page 206, is actually the author's description of the ground-breaking work of Vojta and Faltings. However, the reviewer feels it provides a good summary of the book as a whole, although one might want to add "modular functions and curves" to the list of theories being unified.)
Some readers may find the author's characteristically informal writing style unsettling, especially in an encyclopedia, while others (including this reviewer) will undoubtedly enjoy the pithy comments that save the text from the dullness often found in mathematical compendiums. A typical example, from page 15, reads: "Instead of pseudo-canonical, a variety has been called general type, but with the support of Griffiths, I am trying to make the terminology functional with respect to ideas. (I know I am fighting an uphill battle on this.)"

The book consists, in the main, of definitions, statements of theorems and conjectures, discussions of how the various theorems and conjectures are interrelated, examples, and proof sketches for some of the major results. The author has provided a great service by collecting all of this material in one place, and researchers and graduate students alike will appreciate the extensive index and $400+$ entry bibliography. Further, the author is careful to provide references for all of the proofs that he is forced to omit, although in many cases the primary reference he gives could have been replaced by referring to one of his other works, especially his quartet of basic reference books \ref[ Elliptic curves: Diophantine analysis, Springer, Berlin, 1978; MR 81b:10009; \cit 85j:11005\endcit Fundamentals of Diophantine geometry, Springer, New York, 1983; MR 85j:11005; \cit 88f:32065\endcit Introduction to complex hyperbolic spaces, Springer, New York, 1987; MR 88f:32065; \cit 89m:11059\endcit Introduction to Arakelov theory, Springer, New York, 1988; MR 89m:11059].

The breadth of the material covered is indicated by the following list of chapter titles: I. Some qualitative Diophantine statements. II. Heights and rational points. III. Abelian varieties. IV. Faltings' finiteness theorems on abelian varieties and curves. V. Modular curves over $\bold Q$. VI. The geometric case of Mordell's conjecture. VII. Arakelov theory. VIII. Diophantine problems and complex geometry. IX. Weil functions, integral points and Diophantine approximation. X. Existence of (many) rational points.

This list makes it clear that it would be impossible in a short review to even mention all of the major topics discussed in the book, so we will not try. Suffice it to say that the reviewer feels this book will become a standard reference in the field of Diophantine geometry. Its major failing, for which the author cannot be blamed, is that it is likely to become out-of-date fairly quickly due to the rapid progress being made in the subject. One hopes that the author will provide updated editions every few years. But in any case, there is nothing comparable available today. Every mathematician seriously interested in Diophantine geometry and its burgeoning interrelationships with other fields will certainly want a copy.

\edref{The Russian version has not been received by MR \ref[{\cyr Sovremennye problemy matematiki. Fundamental\cprime nye napravleniya, Tom 60} ( Current problems in mathematics. Fundamental directions, Vol.\ 60), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow].}

Reviewed by Joseph H. Silverman