This remarkable book is an introduction to most of the key areas of diophantine geometry and diophantine approximations (with a notable exception of Baker’s method, which would merit a book on its own). The authors do not strive for utmost generality; they give a clear and thorough account of the principal results but they also include a wide range of additional material, some of which has not appeared in book form before. The first seven chapters develop the classical theory of heights and its applications to Diophantine approximations (the unit equation, Roth’s theorem and Schmidt’s subspace theorem) and to diophantine properties of subvarieties of tori. Chapters 8 to 11 treat geometry and arithmetic of Abelian varieties and Jacobians; they culminate in Bombieri’s version of Vojta’s

proof of Faltings’ theorem (the Mordell conjecture). Chapter 12 discusses the abc-conjecture in its various forms and its consequences. Nevanlinna’s theory is introduced in chapter 13

as a prelude to Vojta’s conjectures in the concluding chapter 14. Three appendices contain, respectively, a thorough summary of algebraic geometry used in the main text (in the language of varieties), ramification theory for number fields and curves, and geometry of numbers.

Reviewer:

jnek