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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
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