Lecture notes on Diophantine analysis. With an appendix by Francesco Amoroso. (English) Zbl 1186.11001
Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) 8. Pisa: Edizioni della Normale (ISBN 978-88-7642-341-3/pbk; 978-88-7642-517-2/ebook). xvi, 237 p. (2009).
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.
The first section deals with some classical diophantine equations, starting with a single variable; next linear equations in two variables and Euclid’s algorithm are discussed; Diophantine approximation including Dirichlet’s lemma are introduced, with application to Pell’s equation and units in quadratic fields; finally, the solution by Gauss and Lagrange to quadratic equations in integers is given.
The second chapter deals with the Thue equation and rational approximation to algebraic numbers; it includes a detailed analysis of the proof of the approximation theorem of Thue.
The study of diophantine equations over algebraic number fields makes it necessary to introduce heights. Chapter 3 discusses the product formula, Weil’s height and Mahler’s measure, the theorems of Northcott and Kronecker, generalisations of Roth’s theorem by Ridout, Roth–Lang, Mahler, the \(S\)-unit equation and Siegel finiteness theorem. It concludes with the study of heights on finitely generated subgroups of \(\mathbb G_m^n\).
The study of heights on subvarieties of \(\mathbb G_m^n\) is the subject of the next chapter. Starting with a problem of Lang, the author discusses lattices and algebraic subgroups, torsion cosets; he gives the theorem of Shou–Wu Zhang as well as the equidistribution result due to Bilu.
The last section is devoted to the \(S\)-unit equation, with a quantitative \(S\)-unit theorem. It concludes with Padé approximation.
An appendix by F. Amoroso includes further results on the Manin–Mumford conjecture (results of Raynaud, Laurent and Hindry), the results of Dobrowolski and Dvornicich on Lehmer’s problem and small height problems.
Each section is supplemented by many remarks, comments, exercices, notes, which provide a lot of further information on these topics, including recent developments and open problem. Hence this booklet starts at an elementary level, including all material which is necessary to understand the subject, and reaches the limit of our current knowledge on all these topics. The large variety of subjects which is involved make it a useful addition to the literature on this fashionable topic, both for the beginner and for the researcher. The wide knowledge of the author, who is one of the noted specialists of the subject, makes from these lecture notes an invaluable reference on these topics.
The first section deals with some classical diophantine equations, starting with a single variable; next linear equations in two variables and Euclid’s algorithm are discussed; Diophantine approximation including Dirichlet’s lemma are introduced, with application to Pell’s equation and units in quadratic fields; finally, the solution by Gauss and Lagrange to quadratic equations in integers is given.
The second chapter deals with the Thue equation and rational approximation to algebraic numbers; it includes a detailed analysis of the proof of the approximation theorem of Thue.
The study of diophantine equations over algebraic number fields makes it necessary to introduce heights. Chapter 3 discusses the product formula, Weil’s height and Mahler’s measure, the theorems of Northcott and Kronecker, generalisations of Roth’s theorem by Ridout, Roth–Lang, Mahler, the \(S\)-unit equation and Siegel finiteness theorem. It concludes with the study of heights on finitely generated subgroups of \(\mathbb G_m^n\).
The study of heights on subvarieties of \(\mathbb G_m^n\) is the subject of the next chapter. Starting with a problem of Lang, the author discusses lattices and algebraic subgroups, torsion cosets; he gives the theorem of Shou–Wu Zhang as well as the equidistribution result due to Bilu.
The last section is devoted to the \(S\)-unit equation, with a quantitative \(S\)-unit theorem. It concludes with Padé approximation.
An appendix by F. Amoroso includes further results on the Manin–Mumford conjecture (results of Raynaud, Laurent and Hindry), the results of Dobrowolski and Dvornicich on Lehmer’s problem and small height problems.
Each section is supplemented by many remarks, comments, exercices, notes, which provide a lot of further information on these topics, including recent developments and open problem. Hence this booklet starts at an elementary level, including all material which is necessary to understand the subject, and reaches the limit of our current knowledge on all these topics. The large variety of subjects which is involved make it a useful addition to the literature on this fashionable topic, both for the beginner and for the researcher. The wide knowledge of the author, who is one of the noted specialists of the subject, makes from these lecture notes an invaluable reference on these topics.
Reviewer: Michel Waldschmidt (Paris)
MSC:
| 11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |
| 11D04 | Linear Diophantine equations |
| 11D09 | Quadratic and bilinear Diophantine equations |
| 11D41 | Higher degree equations; Fermat’s equation |
| 11D59 | Thue-Mahler equations |
| 11G05 | Elliptic curves over global fields |
| 11Jxx | Diophantine approximation, transcendental number theory |
