Lecture Notes on Diophantine Analysis (Publications of the Scuola Normale Superiore Book 8)

These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several supplements, hinted exercises and an appendix on recent work on heights. Cover 1 Title Page 4 Copyright Page 5 Table of Contents 6 Preface 9 Preface to the revised version 9 Notations and conventions 10 Introduction 12 Chapter 1 Some classical diophantine examples 16 1.1. The case of a single variable 16 1.2. The linear case in two variables 17 1.3. Diophantine Approximation 19 1.4. Pell Equation 23 1.4.1. Structure of the solutions and units in quadratic fields 26 1.4.2. Effective solution of Pell and related equations 29 1.5. The general case of degree 2 34 Supplements to Chapter 1 38 Two applications of Dirichlet Lemma 38 First application: Integer solutions of a2 + b2 = p 38 Second application: A factorization algorithm 39 A cyclotomic solution of certain Pell equations 40 A Pell Equation in polynomials 41 Padé Approximations to exp(x) and celebrated irrationalities 43 Rational points on conics 45 A theorem of Fermat 46 Notes to Chapter 1 48 Chapter 2 Thue’s equations and rational approximations 51 2.1. Thue Equations 51 A simple application 53 Relations with Diophantine Approximation 54 2.2. Rational approximations to algebraic numbers 56 Theorem 2.4 implies Theorem 2.1 58 Exponent of approximation 59 2.3. Thue’s method and later developements 60 2.3.1. A rough sketch of Thue’s proof 60 A gap principle 60 Construction of new approximations from a given one 61 Conclusion of the proof 61 A crucial difficulty 61 Precursors of Thue’s method 62 2.3.2. A reformulation and some later refinements 62 Improvements of Thue’s result 64 2.4. Proof of Thue’s Approximation Theorem 65 2.4.1. Preliminaries 65 Differential operators 65 Norms of polynomials 65 Further conventions 65 2.4.2. Construction of polynomials Fn 67 2.4.3. Upper bound for |Dj Fn(u, v)| 70 2.4.4. Lower bound for |Di Fn(u, v)|. 71 2.4.5. An upper bound for the multiplicity at (u, v) 72 2.4.6. Conclusions 74 Another description of the method 77 Supplements to Chapter 2 77 Finiteness of integral points on certain curves 77 Effective decision for an infinity of integral points in genus zero 83 A theorem of Runge 83 A Thue Equation in polynomials 87 Notes to Chapter 2 88 Chapter 3 Heights and diophantine equations over number fields 91 3.1. Fields with a product formula 92 3.1.1. Valuations and the product formula 92 Absolute values 92 Product formula ([77]) 93 3.1.2. Finite extensions 95 3.2. Heights 97 3.2.1. Weil height 97 3.2.2. Mahler’s measure 108 3.2.3. Further properties of the height on Q 111 3.3. Some diophantine analysis over number fields 116 3.3.1. A generalized Roth Theorem 116 3.3.2. S-integers, S-units 118 Heights of S-integers 120 3.3.3. Some diophantine applications 121 3.4. Heights on finitely generated subgroups of Gnm 129 A norm on Zr 131 Extending the norm to Qr 132 Extending the norm to Rr 132 Supplements to Chapter 3 135 The S-unit equation over function fields 135 A different proof and a generalization 137 Detecting multiplicative dependence in Q 141 Specializations preserving multiplicative independence 144 Notes to Chapter 3 146 Chapter 4 Heights on subvarieties of Gnm 149 4.1. A problem of Lang 149 4.2. Lattices and algebraic subgroups 154 4.2.1. Lattices in Zn 154 4.2.2. Algebraic subgroups 156 4.2.3. Some definitions 156 4.2.4. A characterization of torsion cosets 160 Torsion points in algebraic cosets 161 The multiplication maps [m] 161 4.3. Heights on subvarieties of Gnm 164 4.3.1. The theorem of Zhang 164 4.3.2. Bilu’s approach through equidistribution 173 Sketch of deduction of Zhang Theorem from Theorem 4.17 173 Sketch of proof of Theorem 4.17 175 4.4. An application to the S-unit equation 176 Supplements to Chapter 4 181 Lattices and closed subgroups of Rn 181 Discrete subgroups of Rn 181 Closed subgroups of Rn 183 The Skolem-Mahler-Lech Theorem and a generalization 185 An application to Thue Equations 188 A generalization to algebraic groups 188 An open question 190 Notes to Chapter 4 190 Chapter 5 The S-unit equation 193 5.1. A quantitative S-unit theorem 193 5.2. Padé approximations 195 5.3. Proof of Theorem 5.1 199 Plan of the proof 199 5.3.1. Distribution of solutions in euclidean spaces 200 5.3.2. Final arguments 203 Intermediate conclusion 203 5.4. An application 206 Notes to Chapter 5 208 References 210 Index 217 Appendix A Lower bounds for the height (by Francesco Amoroso) 219 A.1. Introduction 219 A.2. Algebraic numbers 220 A.2.1. Sketch of the proof of Theorem A.3 223 A.2.2. Height in Abelian extensions 227 A.2.3. Sketch of proof of Theorem A.4 229 A.3. Subvarieties of Gnm 232 A.3.1. Heights of subvarieties 233 A.3.2. Small height problems 237 References 243 Front Matter....Pages i-xvi Some classical diophantine examples....Pages 1-35 Thue’s equations and rational approximations....Pages 37-76 Heights and diophantine equations over number fields....Pages 77-134 Heights on subvarieties of G m n ....Pages 135-178 The S-unit equation....Pages 179-195 Back Matter....Pages 197-237 "These lecture notes originate from a course covered at the Scuola Normale in Pisa in 2006. The book deals mainly with Diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables."--Jacket Contains lecture notes that originated from a course delivered at the Scuola Normale in Pisa in 2006. This work deals with diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables.