HEIGHTS IN DIOPHANTINE GEOMETRY
CONTENTS 1 HEIGHTS 11 INTRODUCTION 12 ABSOLUTE VALUES 13 FINITE-DIMENSIONAL EXTENSIONS 14 THE PRODUCT FORMULA 15 HEIGHTS IN PROJECTIVE AND AF?NE SPACE 16 HEIGHTS OF POLYNOMIALS 17 LOWER BOUNDS FOR NORMS OF PRODUCTS OF POLYNOMIALS 18 BIBLIOGRAPHICAL NOTES 2 WEIL HEIGHTS 21 INTRODUCTION 22 LOCAL HEIGHTS 23 GLOBAL HEIGHTS 24 WEIL HEIGHTS 25 EXPLICIT BOUNDS FOR WEIL HEIGHTS 26 BOUNDED SUBSETS 27 METRIZED LINE BUNDLES AND LOCAL HEIGHTS 28 HEIGHTS ON GRASSMANNIANS 29 SIEGEL’S LEMMA 210 BIBLIOGRAPHICAL NOTES 3 LINEAR TORI 31 INTRODUCTION 32 SUBGROUPS AND LATTICES 33 SUBVARIETIES AND MAXIMAL SUBGROUPS 34 BIBLIOGRAPHICAL NOTES 4 SMALL POINTS 41 INTRODUCTION 42 ZHANG’S THEOREM 43 THE EQUIDISTRIBUTION THEOREM 44 DOBROWOLSKI’S THEOREM 45 REMARKS ON THE NORTHCOTT PROPERTY 46 REMARKS ON THE BOGOMOLOV PROPERTY 47 BIBLIOGRAPHICAL NOTES 5 THE UNIT EQUATION 51 INTRODUCTION 52 THE NUMBER OF SOLUTIONS OF THE UNIT EQUATION 53 APPLICATIONS 54 EFFECTIVE METHODS 55 BIBLIOGRAPHICAL NOTES 6 ROTH’S THEOREM 61 INTRODUCTION 62 ROTH’S THEOREM 63 PRELIMINARY LEMMAS 64 PROOF OF ROTH’S THEOREM 65 FURTHER RESULTS 66 BIBLIOGRAPHICAL NOTES 7 THE SUBSPACE THEOREM 71 INTRODUCTION 72 THE SUBSPACE THEOREM 73 APPLICATIONS 74 THE GENERALIZED UNIT EQUATION 75 PROOF OF THE SUBSPACE THEOREM 76 FURTHER RESULTS: THE PRODUCT THEOREM 77 THE ABSOLUTE SUBSPACE THEOREM AND THE FALTINGS–W¨USTHOLZ THEOREM 78 BIBLIOGRAPHICAL NOTES 8 ABELIAN VARIETIES 81 INTRODUCTION 82 GROUP VARIETIES 83 ELLIPTIC CURVES 84 THE PICARD VARIETY 85 THE THEOREM OF THE SQUARE AND THE DUAL ABELIAN VARIETY 86 THE THEOREM OF THE CUBE 87 THE ISOGENY MULTIPLICATION BY N 88 CHARACTERIZATION OF ODD ELEMENTS IN THE PICARD GROUP 89 DECOMPOSITION INTO SIMPLE ABELIAN VARIETIES 810 CURVES AND JACOBIANS 811 BIBLIOGRAPHICAL NOTES 9 N´ERON–TATE HEIGHTS 91 INTRODUCTION 92 N´ERON–TATE HEIGHTS 93 THE ASSOCIATED BILINEAR FORM 94 N´ERON–TATE HEIGHTS ON JACOBIANS 95 THE N´ERON SYMBOL 96 HILBERT’S IRREDUCIBILITY THEOREM 97 BIBLIOGRAPHICAL NOTES 10 THE MORDELL–WEIL THEOREM 101 INTRODUCTION 102 THE WEAK MORDELL–WEIL THEOREM FOR ELLIPTIC CURVES 103 THE CHEVALLEY–WEIL THEOREM 104 THE WEAK MORDELL–WEIL THEOREM FOR ABELIAN VARIETIES 105 KUMMER THEORY AND GALOIS COHOMOLOGY 106 THE MORDELL–WEIL THEOREM 107 BIBLIOGRAPHICAL NOTES 11 FALTINGS’S THEOREM 111 INTRODUCTION 112 THE VOJTA DIVISOR 113 MUMFORD’S METHOD AND AN UPPER BOUND FOR THE HEIGHT 114 THE LOCAL EISENSTEIN THEOREM 115 POWER SERIES, NORMS, AND THE LOCAL EISENSTEIN THEOREM 116 A LOWER BOUND FOR THE HEIGHT 117 CONSTRUCTION OF A VOJTA DIVISOR OF SMALL HEIGHT 118 APPLICATION OF ROTH’S LEMMA 119 PROOF OF FALTINGS’S THEOREM 1110 SOME FURTHER DEVELOPMENTS 1111 BIBLIOGRAPHICAL NOTES 12 THE ABC-CONJECTURE 121 INTRODUCTION 122 THE ABC-CONJECTURE 123 BELY?I’S THEOREM 124 EXAMPLES 125 EQUIVALENT CONJECTURES 126 THE GENERALIZED FERMAT EQUATION 127 BIBLIOGRAPHICAL NOTES 13 NEVANLINNA THEORY 131 INTRODUCTION 132 NEVANLINNA THEORY IN ONE VARIABLE 133 VARIATIONS ON A THEME: THE AHLFORS–SHIMIZU CHARACTERISTIC 134 HOLOMORPHIC CURVES IN NEVANLINNA THEORY 135 BIBLIOGRAPHICAL NOTES 14 THE VOJTA CONJECTURES 141 INTRODUCTION 142 THE VOJTA DICTIONARY 143 VOJTA’S CONJECTURES 144 A GENERAL ABC-CONJECTURE 145 THE ABC-THEOREM FOR FUNCTION ?ELDS 146 BIBLIOGRAPHICAL NOTES APPENDIX A ALGEBRAIC GEOMETRY A1 INTRODUCTION A2 AF?NE VARIETIES A3 TOPOLOGY AND SHEAVES A4 VARIETIES A5 VECTOR BUNDLES A6 PROJECTIVE VARIETIES A7 SMOOTH VARIETIES A8 DIVISORS A9 INTERSECTION THEORY OF DIVISORS A10 COHOMOLOGY OF SHEAVES A11 RATIONAL MAPS A12 PROPERTIES OF MORPHISMS A13 CURVES AND SURFACES A14 CONNEXION TO COMPLEX MANIFOLDS APPENDIX B RAMI?CATION B1 DISCRIMINANTS B2 UNRAMI?ED ?ELD EXTENSIONS B3 UNRAMI?ED MORPHISMS B4 THE RAMI?CATION DIVISOR APPENDIX C GEOMETRY OF NUMBERS C1 ADELES C2 MINKOWSKI’S SECOND THEOREM C3 CUBE SLICING REFERENCES GLOSSARY OF NOTATION