Logarithmic Forms and Diophantine Geometry

Baker A., Wustholz G.0511376847, 0521882680, 9780521882682

There is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed, for instance, on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the AndrГc-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Series-title……Page 4
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface……Page 11
1.1 Liouville’s theorem……Page 13
1.2 The Hermite–Lindemann theorem……Page 17
1.3 The Siegel–Shidlovsky theory……Page 21
1.4 Siegel’s lemma……Page 25
1.5 Mahler’s method……Page 28
1.6 Riemann hypothesis over finite fields……Page 32
2.1 Hilbert’s seventh problem……Page 36
2.2 The Gelfond–Schneider theorem……Page 37
2.3 The Schneider–Lang theorem……Page 40
2.4 Baker’s theorem……Page 44
2.5 The Delta-functions……Page 45
2.6 The auxiliary function……Page 48
2.7 Extrapolation……Page 51
2.8 State of the art……Page 53
3.1 Class numbers……Page 58
3.2 The unit equations……Page 61
3.3 The Thue equation……Page 64
3.4 Diophantine curves……Page 66
3.5 Practical computations……Page 69
3.6 Exponential equations……Page 73
3.7 The abc-conjecture……Page 78
4.1 Introduction……Page 82
4.2 Basic concepts in algebraic geometry……Page 85
4.3 The groups Ga and Gm……Page 86
4.4 The Lie algebra……Page 88
4.5 Characters……Page 90
4.6 Subgroup varieties……Page 92
4.7 Geometry of Numbers……Page 94
5.1 Hilbert functions in degree theory……Page 101
5.2 Differential length……Page 105
5.3 Algebraic degree theory……Page 107
5.4 Calculation of the Jacobi rank……Page 109
5.5 The Wüstholz theory……Page 113
5.6 Algebraic subgroups of the torus……Page 118
6.1 Introduction……Page 121
6.2 New applications……Page 129
6.3 Transcendence properties of rational integrals……Page 136
6.4 Algebraic groups and Lie groups……Page 140
6.5 Lindemann’s theorem for abelian varieties……Page 143
6.6 Proof of the integral theorem……Page 147
6.7 Extended multiplicity estimates……Page 148
6.8 Proof of the analytic subgroup theorem……Page 152
6.9 Effective constructions on group varieties……Page 157
7.1 Introduction……Page 161
7.2 Sharp estimates for logarithmic forms……Page 162
7.3 Analogues for algebraic groups……Page 166
7.4 Isogeny theorems……Page 170
7.5 Discriminants, polarisations and Galois groups……Page 174
7.6 The Mordell and Tate conjectures……Page 177
8.2 The Schmidt subspace theorem……Page 179
8.3 Faltings’ product theorem……Page 182
8.4 The André–Oort conjecture……Page 183
8.5 Hypergeometric functions……Page 185
8.6 The Manin–Mumford conjecture……Page 188
References……Page 190
Index……Page 206