Solving polynomial equations in integer or rational numbers is a classical arithmetic problem that can be traced back to the Babylonians but that was studied systematically by Diophantus of Alexandria (ca. 250 AD). Determining whether a given set of polynomial equations has solutions in integer or rational numbers, or finding all such solutions, or counting the number of solutions, are called Diophantine problems. Since a set of polynomial equations defines an affine or projective variety (e.g., over the field of complex numbers) the integer or rational solutions are points on this variety and the language of algebraic geometry gives an appropriate setting to formulate and study these problems, hence the name Diophantine geometry, which accurately describes this feature.
Zannier’s book is a self-contained introduction to some topics in Diophantine geometry. It starts with a preliminary chapter giving examples of classical Diophantine problems, especially “Pell’s equation,” treated using Dirichlet’s theorem on the approximation of irrational algebraic numbers by rational numbers, and sets the stage for the next chapters, where generalizations of this theorem by Liouville, Thue and Roth are proved or quoted and their application to more subtle Diophantine problems such as the S-unit equation or the Mahler-Thue equation are considered. This requires reviewing facts about heights of points in projective space over a field with a product formula. An important consequence is a special case of Siegel’s theorem on the finiteness of the number of integer points on a hyperelliptic curve.
Chapter 4 has new material not treated in any other book on Diophantine geometry, with the exception of the recent book by Bombieri and Gubler that considers some of these topics. Starting by studying the heights of points in powers of the algebraic multiplicative group Gm over an algebraic closure of the rational field Q, the author studies torsion points of small height on algebraic subvarieties X of this power of Gm, giving two elementary proofs of Zhang’s theorem that there is a lower bound for the heights of points on X away from a finite union of translates by torsion points of certain algebraic subgroups of a power of Gm. Zhang’s theorem is then applied to obtain a uniform estimation on the number of solutions of small height of the S-unit equation.
An appendix by F. Amoroso summarizes recent work on lowest bounds for the absolute height of algebraic numbers by Dobrowolski and Amoroso-Dvornicich for algebraic numbers that are not roots of unity in abelian extensions.
Zannier’s book is a very nice introduction to Diophantine geometry, including plenty of exercises dispersed along the text, most with hints for their solution with the aim of keeping the text self-contained. Each chapter ends with a section on notes pointing to further developments and historical comments. The book could be used as a text for a graduate class or for interested students learning this material on their own.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.
1. Classical Diophantine Equations: linear and quadratic equations, Pell Equation, Diophantine Approximation, congruences. Supplements on Pell equations and irrationality of exp(n) and pi. Notes.- 2. Thue's theorems on Diophantine Equations and rational approximations: Description of strategy and detailed proofs. Later refinements. Supplements on integral points on curves and Runge’s theorem. Notes.- 3. Heights and Diophantine equations over number fields: Product formulas, Weil and Mahler heights, Diophantine approximation in number fields, the S-unit equation and its applications. Supplements on the abc-theorem in function fields and on multiplicative dependence of algebraic functions and their values. Notes.- 4. Heights on subvarieties of G_m^n: Torsion points on plane curves and algebraic points of small height on subvarieties of G_m^n. Structure of algebraic subgroups. Theorems of Zhang and Bilu and applications to the S-unit equation. Supplements on discrete and closed subgroups of R^n and on the Skolem-Mahler-Lech theorem. Notes.- 5. The S-unit equation. A sharp quantitative S-unit theorem; explicit Pade’ approximations and the counting of large solutions; counting of small solutions. Applications of the quantitative S-unit theorem. Notes.- Appendix by F. Amoroso: Bounds for the height: Generalized Lehmer problem, Dobrowolski lower bounds. Heights of varieties and extensions of lower bounds to higher dimensions; sharp quantitative Zhang’s theorem.