Lecture Notes on Diophantine Analysis

From the reviews: "This set of lectures originates from an introductory course given by the author at the Scuola Normale Superiore Pisa in 2006-2007 on Diophantine analysis. ... provide a lot of further information on these topics, including recent developments and open problem. Hence this booklet starts at an elementary level ... and reaches the limit of our current knowledge on all these topics. ... a useful addition to the literature on this fashionable topic, both for the beginner and for the researcher." (Michel Waldschmidt, Zentralblatt MATH, Vol. 1186, 2010)

• 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.