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El. knyga: Nevanlinna Theory And Its Relation To Diophantine Approximation (Second Edition)

Min Ru (Univ Of Houston, Usa)

Formatas: 444 pages
Išleidimo metai: 10-Mar-2021
Leidėjas: World Scientific Publishing Co Pte Ltd
Kalba: eng
ISBN-13: 9789811233524

Kitos knygos pagal šią temą:

Complex analysis, complex variables

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Formatas: 444 pages
Išleidimo metai: 10-Mar-2021
Leidėjas: World Scientific Publishing Co Pte Ltd
Kalba: eng
ISBN-13: 9789811233524

Kitos knygos pagal šią temą:

Complex analysis, complex variables

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This book describes the theories and developments in Nevanlinna theory and Diophantine approximation. Although these two subjects belong to the different areas: one in complex analysis and one in number theory, it has been discovered that a number of striking similarities exist between these two subjects. A growing understanding of these connections has led to significant advances in both fields. Outstanding conjectures from decades ago are being solved. Over the past 20 years since the first edition appeared, there have been many new and significant developments. The new edition greatly expands the materials. In addition, three new chapters were added. In particular, the theory of algebraic curves, as well as the algebraic hyperbolicity, which provided the motivation for the Nevanlinna theory.

Preface

vii

1 The Theory of Algebraic Curves

(24)

Part A Nevanlinna Theory

(21)

A1.1 The Riemann-Hurwitz theorem and its consequences

(7)

A1.2 The subdomain case

(5)

A1.3 The theory of algebraic curves in Pn

(9)

Part B Diophantine Approximation

(3)

B1.1 Schmidt's subspace theorem over function fields

(3)

2 The First Main Theorem and the Theory of Height

(48)

Part A Nevanlinna Theory

(29)

A2.1 Sheaves, divisors, line bundles

(21)

A2.2 The Green-Jensen formula

(3)

A2.3 The First Main Theorem

(5)

Part B Diophantine Approximation

(18)

B2.1 The valuation theory

(5)

B2.2 The height and Weil function

(13)

The Correspondence Table

(1)

3 Nevanlinna Theory for Meromorphic Functions and Roth's Theorem

(42)

Part A Nevanlinna Theory

(20)

A3.1 The First Main Theorem

(5)

A3.2 The Logarithmic Derivative Lemma

(10)

A3.3 The Second Main Theorem for meromorphic functions

(4)

Part B Diophantine Approximation

(21)

B3.1 Introduction to Diophantine approximation

(2)

B3.2 Roth's theorem and Vojta's dictionary

(5)

B3.3 Proof of Roth's theorem

100

(14)

The Correspondence Table

114

(1)

4 Holomorphic Curves into Compact Riemann Surfaces

115

(44)

Part A Nevanlinna Theory

115

(19)

A4.1 The Ahlfors-Schwarz Lemma

115

(4)

A4.2 Holomorphic curves into compact Riemann surfaces

119

(8)

A4.3 A new proof of the Logarithmic Derivative Lemma

127

(2)

A4.4 The equi-dimensional theory

129

(5)

Part B Diophantine Approximation

134

(23)

B4.1 Integral points on algebraic curves

134

(1)

B4.2 Curves of genus 0

134

(1)

B4.3 Rational points on curves of genus 1, the Mordell-Weil theorem

135

(17)

B4.4 Integral points on curves of genus 1, the Siegel's theorem

152

(3)

B4.5 Curves of genus greater than or equal to two, the theorem of Faltings

155

(2)

The Correspondence Table

157

(2)

5 Holomorphic Curves in Pn(C) and Schmidt's Subspace Theorem

159

(66)

Part A Nevanlinna Theory

159

(49)

A5.1 Cartan's Second Main Theorem

159

(9)

A5.2 The use of the Second Main Theorem with truncated counting functions

168

(12)

A5.3 Borel's Lemma and its applications

180

(6)

A5.4 The linearly degenerated case

186

(9)

A5.5 Ahlfors' approach

195

(13)

Part B Diophantine Approximation

208

(16)

B5.1 Schmidt's subspace theorem

208

(4)

B5.2 The abc-conjecture

212

(2)

B5.3 The 5-unit lemma

214

(2)

B5.4 The degenerated Schmidt's subspace theorem

216

(8)

The Correspondence Table

224

(1)

6 The Moving Target Problems

225

(38)

Part A Nevanlinna Theory

225

(20)

A6.1 The moving target problem for meromorphic functions

225

(2)

A6.2 The moving target problem for holomorphic curves in projective spaces

227

(7)

A6.3 Cartan's conjecture with moving targets

234

(6)

A6.4 Truncated Second Main Theorem with moving targets

240

(5)

Part B Diophantine Approximation

245

(17)

B6.1 Schmidt's subspace theorem with moving targets

245

(7)

B6.2 The degenerate case

252

(5)

B6.3 Applications of Schmidt's subspace theorem with moving targets

257

(5)

The Correspondence Table

262

(1)

7 Extension of Cartan's Theorem and Schmidt's Subspace Theorem

263

(52)

Part A Nevanlinna Theory

263

(35)

A7.1 The Second Main Theorem for general divisors on projective varieties

263

(7)

A7.2 Results derived by computing the Nevanlinna constant

270

(7)

A7.3 Holomorphic curves intersecting divisors in general and subgeneral positions

277

(21)

Part B Diophantine Approximation

298

(15)

B7.1 The Nevanlinna constant

298

(8)

B7.2 The result of Ru-Vojta

306

(7)

The Correspondence Table

313

(2)

8 Equi-dimensional Nevanlinna Theory and Vojta's Conjecture

315

(18)

Part A Nevanlinna Theory

315

(14)

A8.1 Logarithmic Derivative Lemma for meromorphic functions on Cn

315

(9)

A8.2 The equi-dimensional Nevanlinna theory

324

(4)

A8.3 Griffiths' conjecture

328

(1)

Part B Diophantine Approximation

329

(3)

B8.1 Vojta's conjecture in Diophantine approximation

329

(3)

The Correspondence Table

332

(1)

9 Holomorphic Curves in Abelian Varieties and the Theorem of Faltings

333

(26)

Part A Nevanlinna Theory

333

(22)

A9.1 Bloch's theorem for holomorphic curves in Abelian varieties

333

(14)

A9.2 The Second Main Theorem for holomorphic curves into abelian varieties

347

(3)

A9.3 McQuillan's proof

350

(5)

Part B Diophantine Approximation

355

(2)

B9.1 Faltings' Theorem on rational points in abelian varieties

355

(2)

The Correspondence Table

357

(2)

10 Complex Hyperbolic Manifolds and Lang's Conjecture

359

(54)

Part A Nevanlinna Theory

359

(52)

A10.1 The Schwarz lemma

359

(3)

A10.2 Kobayashi hyperbolicity

362

(7)

A10.3 Brody's hyperbolicity

369

(4)

A10.4 Algebraic hyperbolicity

373

(8)

A10.5 Differential geometric criteria for hyperbolicity

381

(12)

A10.6 The construction of the Finsler metric

393

(11)

A10.7 Jet differentials and the fundamental vanishing theorem

404

(7)

Part B Diophantine Approximation

411

(2)

B10.1 Lang's conjecture

411

(2)

Bibliography

413

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