Value Distribution Theory Related to Number Theory

Contents

Preface

Heights

1.1 Field extensions

1.2 Fields with valuations

1.3 Discriminant of field extensions

1.4 Product formula

1.5 Hermitian geometry

1.6 Basic geometric notions

1.7 Weil functions

1.8 Heights in number fields

1.9 Functorial properties of heights

1.10 Gauss’ lemma

Nevanlinna Theory

2.1 Notions in complex geometry

2.2 Kobayashi hyperbolicity

2.3 Characteristic functions

2.4 Growth of rational functions

2.5 Lemma of the logarithmic derivative

2.6 Second main theorem

2.7 Notes on the second main theorem

2.8 The Cartan-Nochka theorem

2.9 First main theorem for line bundles

2.10 Jacobian sections

2.11 Stoll’s theorems

2.12 Carlson-Griffiths-King theory

Topics in Number Theory

3.1 Elliptic curves

3.2 The

conjecture

3.3 Mordell’s conjecture and generalizations

3.4 Fermat equations and Waring’s problem

3.5 Thue-Siegel-Roth’s theorem

3.6 Schmidt’s subspace theorem

3.7 Vojta’s conjectures

3.8 Subspace theorems on hypersurfaces

3.9 Vanishing sums in function fields

Function Solutions of Diophantine Equations

4.1 Nevanlinna’s third main theorem

4.2 Generalized Mason’s theorem

4.3 Generalized

conjecture

4.4 Generalized Hall’s conjecture

4.5 Borel’s theorem and its analogues

4.6 Meromorphic solutions of Fermat equations

4.7 Waring’s problem for meromorphic functions

4.8 Holomorphic curves into a complex torus

4.9 Hyperbolic spaces of lower dimensions

4.10 Factorization of functions

4.11 Wiman-Valiron theory

Functions over Non- Archimedean Fields

5.1 Equidistribution formula

5.2 Second main theorem of meromorphic functions

5.3 Equidistribution formula for hyperplanes

5.4 Non-Archimedean Cartan-Nochka theorem

5.5 Holomorphic curves into projective varieties

5.6 The

theorem for meromorphic functions

5.7 The

theorem for entire functions

5.8 Non-Archimedean Borel theorem

5.9 Waring’s problem over function fields

5.10 Picard-Berkovich’s theorem

Holomorphic Curves in Canonical Varieties

6.1 Variations of the first main theorem

6.2 Meromorphic connections

6.3 Siu theory

6.4 Bloch-Green’s conjecture

6.5 Green-Griffiths’ conjecture

6.6 Notes on Griffiths’ and Lang’s conjectures

Riemann’s

function

7.1 Riemann’s functional equation

7.2 Converse theorems

7.3 Riemann’s hypothesis

7.4 Hadamard’s factorization

7.5 Nevanlinna’s formula

7.6 Carleman’s formula

7.7 Levin’s formula

7.8 Notes on Nevanlinna’s conjecture

Bibliography

Symbols

Index