Spis treści: Lecture Notes on Diophantine Analysis


Preface, s. ix
Introduction, s. xiii

1 Some classical diophantine examples, s. 1

1.1 The case of a single variable, s. 1
1.2 The linear case in two variables, s. 2
1.3 Diophantine Approximation, s. 4
1.4 Pell Equation, s. 7
1.4.1 Structure of the solutions and units in quadratic fields, s. 11
1.4.2 Effective solution of Pell and related equations, s. 14
1.5 The general case of degree 2, s. 19

Supplements, s. 22

Two applications of Dirichlet Lemma, s. 22
A cyclotomic solution of certain Pell equations, s. 24
A Pell Equation in polynomials, s. 25
Pade Approximations to exp(x) and celebrated irrationalities, s. 27
Rational points on conies, s. 28
A theorem of Fermat, s. 29
Notes, s. 30

2 Thue's equations and rational approximations, s. 35

2.1 Thue Equations, s. 35
2.2 Rational approximations to algebraic numbers, s. 39
2.3 Thue's method and later developements, s. 42
2.3.1 A rough sketch of Thue's proof, s. 42
2.3.2 A reformulation and some later refinements, s. 45
2.4 Proof of Thue's Approximation Theorem, s. 47
2.4.1 Preliminaries, s. 47
2.4.2 Construction of polynomials F„, s. 49
2.4.3 Upper bound for |DjFn(u, v)|, s. 2
2.4.4 Lower bound for |DjFn(u, v)|, s. 54
2.4.5 An upper bound for the multiplicity at (u, v), s. 54
2.4.6 Conclusions, s. 56

Supplements, s. 60

Finiteness of integral points on certain curves, s. 60
Effective decision for an infinity of integral points in genus zero, s. 66
A theorem of Runge, s. 66
A polynomial Thue Equation, s. 70
Notes, s. 71

3 Heights and diophantine equations over number fields, s. 75

3.1 Fields with a product formula, s. 76
3.1.1 Valuations and the product formula, s. 76
3.1.2 Finite extensions, s. 79
3.2 Heights, s. 81
3.2.1 Weil height, s. 81
3.2.2 Mahler's measure, s. 92
3.2.3 Further properties of the height on Q, s. 95
3.3 Some diophantine analysis over number fields, s. 100
3.3.1 A generalized Roth Theorem, s. 100
3.3.2 5-integers, S-units, s. 102
3.3.3 Some diophantine applications, s. 105
3.4 Heights on finitely generated subgroups of Gnm, s. 113

Supplements, s. 120

The S-unit equation over function fields, s. 120
Detecting multiplicative dependence in Q, s. 125
Specializations preserving multiplicative independence, s. 128
Notes, s. 130

4 Heights on subvarieties of Gnm, s. 135

4.1 A problem of Lang, s. 135
4.2 Lattices and algebraic subgroups, s. 140
4.2.1 Lattices in Zn, s. 140
4.2.2 Algebraic subgroups, s. 142
4.2.3 Some definitions, s. 142
4.2.4 A characterization of torsion cosets, s. 146
4.3 Heights on subvarieties of Gnm, s. 150
4.3.1 The theorem of Zhang, s. 150
4.3.2   Bilu's approach through equidistribution, s. 159
4.4   An application to the 5-unit equation, s. 162

Supplements, s. 167

Lattices and closed subgroups of Rn, s. 167
The Skolem-Mahler-Lech Theorem and a generalization, s. 171
Notes, s. 176

5   The S-unit equation, s. 179

5.1 A quantitative S-unit theorem, s. 179
5.2 Pade approximations, s. 181
5.3 Proof of Theorem 5.1, s. 185
5.3.1 Distribution of solutions in euclidean spaces, s. 186
5.3.2 Final arguments, s. 189
5.4 An application, s. 192

Notes, s. 194
References, s. 197
Index, s. 205
Appendix, s. 207

A Lower bounds for the height (by Francesco Amoroso), s. 207

A.l  Introduction, s. 207
A.2  Algebraic numbers, s. 208
A.2.1   Sketch of the proof of Theorem A.3, s. 211
A.2.2   Height in Abelian extensions, s. 215
A.2.3   Sketch of proof of Theorem A.4, s. 217
A.3  Subvarieties of Gnm, s. 220
A.3.1   Heights of subvarieties, s. 221
A.3.2   Small height problems, s. 225
References, s. 231

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