Value Distribution Theory Related to Number Theory (eBook)

Contents 4
Preface 8
Heights 11
1.1 Field extensions 11
1.2 Fields with valuations 21
1.3 Discriminant of field extensions 31
1.4 Product formula 43
1.5 Hermitian geometry 46
1.6 Basic geometric notions 60
1.7 Weil functions 79
1.8 Heights in number fields 83
1.9 Functorial properties of heights 88
1.10 Gauss’ lemma 93
Nevanlinna Theory 98
2.1 Notions in complex geometry 98
2.2 Kobayashi hyperbolicity 139
2.3 Characteristic functions 148
2.4 Growth of rational functions 155
2.5 Lemma of the logarithmic derivative 159
2.6 Second main theorem 166
2.7 Notes on the second main theorem 172
2.8 The Cartan-Nochka theorem 176
2.9 First main theorem for line bundles 184
2.10 Jacobian sections 191
2.11 Stoll’s theorems 200
2.12 Carlson-Griffiths-King theory 206
Topics in Number Theory 221
3.1 Elliptic curves 221
3.2 The 240
conjecture 240
3.3 Mordell’s conjecture and generalizations 245
3.4 Fermat equations and Waring’s problem 248
3.5 Thue-Siegel-Roth’s theorem 251
3.6 Schmidt’s subspace theorem 253
3.7 Vojta’s conjectures 257
3.8 Subspace theorems on hypersurfaces 264
3.9 Vanishing sums in function fields 280
Function Solutions of Diophantine Equations 294
4.1 Nevanlinna’s third main theorem 294
4.2 Generalized Mason’s theorem 303
4.3 Generalized 308
conjecture 308
4.4 Generalized Hall’s conjecture 311
4.5 Borel’s theorem and its analogues 315
4.6 Meromorphic solutions of Fermat equations 326
4.7 Waring’s problem for meromorphic functions 335
4.8 Holomorphic curves into a complex torus 343
4.9 Hyperbolic spaces of lower dimensions 348
4.10 Factorization of functions 363
4.11 Wiman-Valiron theory 371
Functions over Non- Archimedean Fields 378
5.1 Equidistribution formula 378
5.2 Second main theorem of meromorphic functions 386
5.3 Equidistribution formula for hyperplanes 391
5.4 Non-Archimedean Cartan-Nochka theorem 396
5.5 Holomorphic curves into projective varieties 401
5.6 The 406
theorem for meromorphic functions 406
5.7 The 410
theorem for entire functions 410
5.8 Non-Archimedean Borel theorem 413
5.9 Waring’s problem over function fields 417
5.10 Picard-Berkovich’s theorem 420
Holomorphic Curves in Canonical Varieties 427
6.1 Variations of the first main theorem 427
6.2 Meromorphic connections 435
6.3 Siu theory 441
6.4 Bloch-Green’s conjecture 450
6.5 Green-Griffiths’ conjecture 453
6.6 Notes on Griffiths’ and Lang’s conjectures 456
Riemann’s 466
function 466
7.1 Riemann’s functional equation 466
7.2 Converse theorems 472
7.3 Riemann’s hypothesis 478
7.4 Hadamard’s factorization 485
7.5 Nevanlinna’s formula 491
7.6 Carleman’s formula 498
7.7 Levin’s formula 506
7.8 Notes on Nevanlinna’s conjecture 509
Bibliography 512
Symbols 533
Index 536