Logarithmic Forms and Diophantine Geometry

by A. Baker , G. Wüstholz

• ISBN13: 9780521882682
• ISBN10: 0521882680
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-02-18
• Publisher: Cambridge University Press
• More Book Details

Summary

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. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, advanced topics including a new, more algebraic, derivation of multiplicity estimates. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the André-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in diophantine geometry.

Table of Contents

Preface	p. ix
Transcendence origins	p. 1
Liouville's theorem	p. 1
The Hermite-Lindemann theorem	p. 5
The Siegel-Shidlovsky theory	p. 9
Siegel's lemma	p. 13
Mahler's method	p. 16
Riemann hypothesis over finite fields	p. 20
Logarithmic forms	p. 24
Hilbert's seventh problem	p. 24
The Gelfond-Schneider theorem	p. 25
The Schneider-Lang theorem	p. 28
Baker's theorem	p. 32
The [Delta]-functions	p. 33
The auxiliary function	p. 36
Extrapolation	p. 39
State of the art	p. 41
Diophantine problems	p. 46
Class numbers	p. 46
The unit equations	p. 49
The Thue equation	p. 52
Diophantine curves	p. 54
Practical computations	p. 57
Exponential equations	p. 61
The abc-conjecture	p. 66
Commutative algebraic groups	p. 70
Introduction	p. 70
Basic concepts in algebraic geometry	p. 73
The groups G[superscript a] and G[superscript m]	p. 74
The Lie algebra	p. 76
Characters	p. 78
Subgroup varieties	p. 80
Geometry of Numbers	p. 82
Multiplicity estimates	p. 89
Hilbert functions in degree theory	p. 89
Differential length	p. 93
Algebraic degree theory	p. 95
Calculation of the Jacobi rank	p. 97
The Wustholz theory	p. 101
Algebraic subgroups of the torus	p. 106
The analytic subgroup theorem	p. 109
Introduction	p. 109
New applications	p. 117
Transcendence properties of rational integrals	p. 124
Algebraic groups and Lie groups	p. 128
Lindemann's theorem for abelian varieties	p. 131
Proof of the integral theorem	p. 135
Extended multiplicity estimates	p. 136
Proof of the analytic subgroup theorem	p. 140
Effective constructions on group varieties	p. 145
The quantitative theory	p. 149
Introduction	p. 149
Sharp estimates for logarithmic forms	p. 150
Analogues for algebraic groups	p. 154
Isogeny theorems	p. 158
Discriminants, polarisations and Galois groups	p. 162
The Mordell and Tate conjectures	p. 165
Further aspects of Diophantine geometry	p. 167
Introduction	p. 167
The Schmidt subspace theorem	p. 167
Faltings' product theorem	p. 170
The Andre-Oort conjecture	p. 171
Hypergeometric functions	p. 173
The Manin-Mumford conjecture	p. 176
References	p. 178
Index	p. 194
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