Logarithmic Forms and Diophantine Geometry

Preface ix Transcendence origins 1(23) Liouville's theorem 1(4) The Hermite--Lindemann theorem 5(4) The Siegel--Shidlovsky theory 9(4) Siegel's lemma 13(3) Mahler's method 16(4) Riemann hypothesis over finite fields 20(4) Logarithmic forms 24(22) Hilbert's seventh problem 24(1) The Gelfond--Schneider theorem 25(3) The Schneider--Lang theorem 28(4) Baker's theorem 32(1) The Δ-functions 33(3) The auxiliary function 36(3) Extrapolation 39(2) State of the art 41(5) Diophantine problems 46(24) Class numbers 46(3) The unit equations 49(3) The Thue equation 52(2) Diophantine curves 54(3) Practical computations 57(4) Exponential equations 61(5) The abc-conjecture 66(4) Commutative algebraic groups 70(19) Introduction 70(3) Basic concepts in algebraic geometry 73(1) The groups Ga and Gm 74(2) The Lie algebra 76(2) Characters 78(2) Subgroup varieties 80(2) Geometry of Numbers 82(7) Multiplicity estimates 89(20) Hilbert functions in degree theory 89(4) Differential length 93(2) Algebraic degree theory 95(2) Calculation of the Jacobi rank 97(4) The Wustholz theory 101(5) Algebraic subgroups of the torus 106(3) The analytic subgroup theorem 109(40) Introduction 109(8) New applications 117(7) Transcendence properties of rational integrals 124(4) Algebraic groups and Lie groups 128(3) Lindemann's theorem for abelian varieties 131(4) Proof of the integral theorem 135(1) Extended multiplicity estimates 136(4) Proof of the analytic subgroup theorem 140(5) Effective constructions on group varieties 145(4) The quantitative theory 149(18) Introduction 149(1) Sharp estimates for logarithmic forms 150(4) Analogues for algebraic groups 154(4) Isogeny theorems 158(4) Discriminants, polarisations and Galois groups 162(3) The Mordell and Tate conjectures 165(2) Further aspects of Diophantine geometry 167(11) Introduction 167(1) The Schmidt subspace theorem 167(3) Faltings' product theorem 170(1) The Andre--Oort conjecture 171(2) Hypergeometric functions 173(3) The Manin--Mumford conjecture 176(2) References 178(16) Index 194