Diophantine geometry : an introduction /
| Main Author: | |
|---|---|
| Corporate Author: | |
| Other Authors: | |
| Format: | eBook |
| Language: | English |
| Published: |
New York :
Springer,
[2000]
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| Series: | Graduate texts in mathematics ;
201. |
| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Part A Geometry of Curves and Abelian Varieties 6
- A.1 Algebraic Varieties 8
- A.2 Divisors 34
- A.3 Linear Systems 49
- A.4 Algebraic Curves 67
- A.5 Abelian Varieties over C 91
- A.6 Jacobians over C 110
- A.7 Abelian Varieties over Arbitrary Fields 119
- A.8 Jacobians over Arbitrary Fields 134
- A.9 Schemes 151
- Part B Height Functions 168
- B.1 Absolute Values 170
- B.2 Heights on Projective Space 174
- B.3 Heights on Varieties 183
- B.4 Canonical Height Functions 195
- B.5 Canonical Heights on Abelian Varieties 199
- B.6 Counting Rational Points on Varieties 210
- B.7 Heights and Polynomials 224
- B.8 Local Height Functions 237
- B.9 Canonical Local Heights on Abelian Varieties 241
- B.10 Introduction to Arakelov Theory 243
- Part C Rational Points on Abelian Varieties 257
- C.1 Weak Mordell-Weil Theorem 260
- C.2 Kernel of Reduction Modulo p 267
- C.3 Appendix Finiteness Theorems in Algebraic Number Theory 273
- C.4 Appendix Selmer and Tate-Shafarevich Groups 279
- C.5 Appendix Galois Cohomology and Homogeneous Spaces 283
- Part D Diophantine Approximation and Integral Points on Curves 299
- D.1 Two Elementary Results on Diophantine Approximation 300
- D.2 Roth's Theorem 304
- D.3 Preliminary Results 307
- D.4 Construction of the Auxiliary Polynomial 316
- D.5 Index Is Large 323
- D.6 Index Is Small (Roth's Lemma) 329
- D.7 Completion of the Proof of Roth's Theorem 341
- D.8 Application: The Unit Equation U + V = 1 345
- D.9 Application: Integer Points on Curves 353
- Part E Rational Points on Curves of Genus at Least 2 367
- E.1 Vojta's Geometric Inequality and Faltings' Theorem 369
- E.2 Pinning Down Some Height Functions 373
- E.3 An Outline of the Proof of Vojta's Inequality 379
- E.4 An Upper Bound for h[subscript Omega](z, w) 381
- E.5 A Lower Bound for h[subscript Omega](z, w) for Nonvanishing Sections 385
- E.6 Constructing Sections of Small Height I: Applying Riemann-Roch 389
- E.7 Constructing Sections of Small Height II: Applying Siegel's Lemma 393
- E.8 Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version I 401
- E.9 Eisenstein's Estimate for the Derivatives of an Algebraic Function 408
- E.10 Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version II 412
- E.11 A Nonvanishing Derivative of Small Order 418
- E.12 Completion of the Proof of Vojta's Inequality 421
- Part F Further Results and Open Problems 433
- F.1 Curves and Abelian Varieties 434
- F.1.1 Rational Points on Subvarieties of Abelian Varieties 434
- F.1.2 Application to Points of Bounded Degree on Curves 439
- F.2 Discreteness of Algebraic Points 443
- F.2.1 Bogomolov's Conjecture 444
- F.2.2 Height of a Variety 445
- F.3 Height Bounds and Height Conjectures 451
- F.4 Search for Effectivity 456
- F.4.1 Effective Computation of the Mordell-Weil Group A([kappa]) 457
- F.4.2 Effective Computation of Rational Points on Curves 465
- F.4.3 Quantitative Bounds for Rational Points 472
- F.5 Geometry Governs Arithmetic 474
- F.5.1 Kodaira Dimension 475
- F.5.2 Bombieri-Lang Conjecture 479
- F.5.3 Vojta's Conjecture 482
- F.5.4 Varieties Whose Rational Points Are Dense 487
- Part A Geometry of Curves and Abelian Varieties 6
- A.1 Algebraic Varieties 8
- A.1.1 Affine and Projective Varieties 9
- A.1.2 Algebraic Maps and Local Rings 15
- A.1.3 Dimension 22
- A.1.4 Tangent Spaces and Differentials 24
- A.2 Divisors 34
- A.2.1 Weil Divisors 34
- A.2.2 Cartier Divisors 37
- A.2.3 Intersection Numbers 44
- A.3 Linear Systems 49
- A.3.1 Linear Systems and Maps 49
- A.3.2 Ampleness and the Enriques-Severi-Zariski Lemma 52
- A.3.3 Line Bundles and Sheavese 56
- A.4 Algebraic Curves 67
- A.4.1 Birational Models of Curves 68
- A.4.2 Genus of a Curve and the Riemann-Roch Theorem 70
- A.4.3 Curves of Genus 0 74
- A.4.4 Curves of Genus 1 76
- A.4.5 Curves of Genus at Least 2 81
- A.4.6 Algebraic Surfaces 84
- A.5 Abelian Varieties over C 91
- A.5.1 Complex Tori 93
- A.5.2 Divisors, Theta Functions, and Riemann Forms 97
- A.5.3 Riemann-Roch for Abelian Varieties 103
- A.6 Jacobians over C 110
- A.6.1 Abelian Integrals 110
- A.6.2 Periods of Riemann Surfaces 111
- A.6.3 Jacobian of a Riemann Surface 113
- A.6.4 Albanese Varieties 116
- A.7 Abelian Varieties over Arbitrary Fields 119
- A.7.1 Generalities 119
- A.7.2 Divisors and the Theorem of the Cube 121
- A.7.3 Dual Abelian Varieties and Poincare Divisors 128
- A.8 Jacobians over Arbitrary Fields 134
- A.8.1 Construction and Properties 134
- A.8.2 Divisor [Theta] 138
- A.8.3 Appendix Families of Subvarieties 142
- A.9 Schemes 151
- A.9.1 Varieties over Z 151
- A.9.2 Analogies Between Number Fields and Function Fields 159
- A.9.3 Minimal Model of a Curve 160
- A.9.4 Neron Model of an Abelian Variety 162.