New Mathematical Monographs 4. Cambridge: Cambridge University Press (ISBN 0-521-84615-3/hbk). xvi, 652 p. £ 75.00; $ 130.00 (2006).

Polynomial equations in integer or rational numbers are certainly amongst the oldest subjects of study in mathematics. For thousands of years, their unique fascination has attracted all kinds of mathematical researchers, ranging from a countless number of ambitious amateurs up to the greatest minds in the history of mathematics. Due to its appealing intrinsic beauty, its intellectually challenging down-to-earth nature, and its numerous spectacular results, conjectures, and still open problems, the theory of Diophantine equations has maintained, over all the centuries, its prominent role as the probably most popular part of number theory. During the last century, the development of modern algebraic geometry and algebraic number theory has led to a tremendous progress in the theory of Diophantine equations. The study of these equations from their underlying algebro-geometric point of view is nowadays called Diophantine geometry or, synonymously, arithmetic (algebraic) geometry, and the recent great results in this (old and new) area of mathematics have been thrilling to such a degree that even daily newspapers reported on them, and made their statements popular worldwide. However, in contrast to the elementary concern of the theory of Diophantine equations, Diophantine geometry is a highly advanced and sophisticated subject in rapid progress, incorporating various parts of modern algebra, algebraic geometry, and complex analysis. Therefore a comprehensive, largely self-contained monograph on both basic and advanced topics in Diophantine geometry would be more than desirable and useful, although its composition would necessarily be a monumental task for any expert author. Indeed, there are already a few excellent books devoted to Diophantine geometry, above all the meanwhile classic standard texts by {\it S. Lang} [Fundamentals of Diophantine Geometry, New York: Springer Verlag (1983;

Zbl 0528.14013)], by {\it J.-P. Serre} [Lectures on the Mordell-Weil Theorem, Braunschweig: Vieweg Verlag (1989;

Zbl 0676.14005)], and by {\it M. Hindry} and {\it J. H. Silverman} [Diophantine Geometry: An Introduction, New York: Springer Verlag (2000;

Zbl 0948.11023)], but most of these texts focus on the recent great finiteness theorems, thereby developing the basic concepts and tools only as far as directly needed. The book under review represents another comprehensive, far-reaching introduction to Diophantine geometry, though by following a different strategy. Namely, the fundamental concepts of height functions in Diophantine geometry, as used as crucial ingredients in the proofs of the various finiteness theorems of Mordell-Weil, Roth, Siegel, Faltings, Fermat-Wiles, and others, are here the central objects of study. Their predominant importance, ubiquity, and universality in contemporary Diophantine geometry is the main theme of the present book, with some of the great finiteness theorems and other related topics serving as effective applications of the analysis of height functions. In this vein, the central role of heights in modern Diophantine geometry is placed in the foreground of the entire treatise, and their geometry is elaborated in unparalleled depth. This fine analysis of height functions transpires a wealth of new viewpoints and insights concerning their crucial significance and the authors’ re-examination of many approaches, methods, and results in Diophantine geometry, under the aspect of height functions, leads to a more systematic conceptual path through the subject for graduate students and beginning researchers. Moreover, the authors have tried to demonstrate the link between classical Diophantine geometry and the modern approach via arithmetic algebraic geometry in a natural way, throughout the entire text, and to keep the presentation of the material as self-contained as possible. Thus, as to the latter claim, the book even contains separate introductory chapters on basic abstract algebraic geometry (as a large appendix), the theory of Abelian varieties, algebraic ramification theory (also as an appendix), and complex value distribution theory (Nevanlinna theory), mainly with full proofs of the central theorems in the respective related topics. This is certainly comprehensiveness in matchless a manner, very much to the benefit of all kinds of readers. The present voluminous book comes with fourteen chapters and three rather large appendices, each of which begins with its own brief introduction. The titles of the single parts of the book read as follows: 1. Heights; 2. Weil Heights; 3. Linear Tori; 4. Small Points; 5. The Unit Equation; 6. Roth’s Theorem; 7. The Subspace Theorem; 8. Abelian Varieties; 9. Neron-Tate Heights; 10. The Mordell-Weil Theorem; 11. Faltings’s Theorem; 12. The abc-Conjecture; 13. Nevanlinna Theory; 14. The Vojta Conjectures; Appendix A: Algebraic Geometry; Appendix B: Ramification; Appendic C: Geometry of Numbers. As for the contents of the single chapters, already the headlines indicate that Chapters 3, 8 and 13 provide the necessary auxiliary materials from the respective areas in algebraic geometry and complex analysis, whereas Appendices A, B and C thoroughly introduce the reader to the classical basics of the respective theories. Chapter 1 contains preliminary standard material on absolute values and the elementary theory of height functions on projective varieties, but also some of the finer results on classical heights which are not usually treated in other textbooks. Chapter 2 studies heights from the abstract algebro-geometric point of view, including local and global Weil heights, Northcott’s famous finiteness theorem, heights on Grassmannians, Siegel’s Lemma, and some new material on explicit bounds for Weil heights. Also, in this chapter, the reader meets metrized line bundles and their significance in the study of local height functions. Chapter 4 provides an excellent introduction to the recently developed theory of small points, that is to the study of the distribution of algebraic points of small height on subvarieties of algebraic tori. Some of the topics treated in this context are Zhang’s theorem, Lehmer’s problem, Dobrowolski’s theorem, and a result of {\it F. Amoroso} and {\it R. Dvornicich} (2000) on a lower bound for heights in Abelian extensions. Recent further-reaching results in this realm are pointed out, and several new enlightening examples enhance the material of this highly topical chapter. Chapter 5 offers a very elegant and up-to-date account of the unit equation, including the recent spectacular work of {\it F. Beukers} and {\it H. P. Schlickewei} (1996) as well as a new approach to Hilbert’s irreducibility theorem via {\it V. G. Sprindzhuk’s} theory of arithmetic specializations in polynomials (1983). Chapter 6 is exclusively devoted to Roth’s theorem about approximations of algebraic numbers by rational numbers, ranging from its classical origins (Thue’s theorem) to {\it P. Vojta’s} recent generalization with moving targets (1996). Chapter 7 deals then with W. M. Schmidt’s far-reaching generalization of Roth’s theorem to systems of inequalities in linear forms, culminating in a thorough analysis of Schmidt’s celebrated subspace theorem (1989) and its important modifications by {\it H. P.Schlickewei} (1992) and by {\it J.-H. Evertse} (1996). Further results in this direction by Faltings-Wüstholz and {\it Evertse-Ferretti} (2002) are briefly explained, without proofs, and some concrete applications are given as well. Chapters 9, 10 and 11 concern Abelian varieties and their subvarieties in Diophantine geometry. The material covered here incorporates Néron-Tate heights (on Jacobians of arithmetic curves), the Néron symbol, the Chevalley-Weil theorem, the weak Mordell-Weil theorem for elliptic curves and for general Abelian varieties, Faltings’s famous theorem establishing Mordell’s conjecture for arithmetic curves (1983), and P. Vojta’s approach as simplified by {\it E. Bombieri} (1990). The final Chapters 12, 13 and 14 turn to the relations between (classical) Diophantine geometry and modern arithmetic geometry, the latter of which would have required a separate monograph by itself and, therefore, could not be an exhaustive part of the present book.
Nevertheless, the authors at times straddle the borderline between these two topics, thereby wetting the appetite for further studies by the reader. Chapter 12 treats the so-called “abc-conjecture” over number fields, including Belyi’s theorem, Elkies’s theorem, a finiteness result for the generalized Fermat equation due to {\it H. Darmon} and {\it A. Granville} (1995), and numerous (partially new) examples. Chapter 13 gives a self-contained introduction to classical Nevanlinna theory and H. Cartan’s extension to the theory of meromorphic curves, which is to motivate the concluding Chapter 14 on the celebrated Vojta conjectures. The Vojta conjectures may be considered as an arithmetic counterpart of classical value distribution theory, and they contain the abc-conjecture as an important special case. Developing Vojta’s dictionary between Nevanlinna theory and the theory of Diophantine approximation [{\it P. Vojta}, Diophantine Approximations and Value Distribution Theory, Lect. Notes Math. 1239, Springer Verlag, (1987;

Zbl 0609.14011)] at the beginning, the authors review the main results of the previous chapters, within this framework, in the sequel, with particular emphasis put on the abc-conjecture in its most general form and, finally, on the abc-theorem for function fields à la {\it J. F. Voloch} (1985) and {\it W. D.Brownawell} and {\it D. W. Masser} (1986). Along the way, the authors also discuss the related theorem of {\it W. W. Stothers} (1981) and {\it R. C. Mason} (1983) on local heights on projective curves over a field. As mentioned before, the three appendices on basic (classical) algebraic geometry, ramification theory, and the geometry of numbers provide full accounts of the necessary prerequisites from these areas of mathematics. There is a nearly exhaustive bibliography with 343 references to books and research articles. This bibliography is not only utmost up-to-date but also abundantly referred to throughout the text. In fact, each chapter ends with a section on bibliographical notes, historical remarks, and concrete hints for further reading,which enhances the value of the present monograph in very exemplary a manner. Also the careful glossary of notation and the complete index at the end of the book are utmost beneficial and helpful to the reader. Some sections in this book appear in small print, indicating that they can be omitted in a first reading, either they require additional knowledge of algebra and geometry, or that they treat some side topics not appearing elsewhere in the text. For all that, these special sections are written in just as complete a manner, and they should be of equal interest to the inquisitive reader because they provide an additional wealth of related information. Finally, the vast amount of illustrating examples in each single chapter represents an extra value of this masterpiece of a mathematical text. Apart from being another great standard work in Diophantine geometry, standing out by both its original approach and its unrivaled comprehensiveness, this book must be seen as a typical example of mastery in mathematical writing, and as an indispensable source for every researcher in the fields of Diophantine geometry, algebraic number theory, and arithmetic algebraic geometry likewise.