Diophantine geometry : an introduction /
Saved in:
| Author / Creator: | Hindry, Marc. |
|---|---|
| Imprint: | New York : Springer, c2000. |
| Description: | xiii, 558 p. : ill. ; 25 cm. |
| Language: | English |
| Series: | Graduate texts in mathematics. 201 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4244876 |
Table of Contents:
- Preface
- Acknowledgments
- Contents
- Detailed Contents for Part A
- Introduction
- Part A. The Geometry of Curves and Abelian Varieties
- A.1. Algebraic Varieties
- A.2. Divisors
- A.3. Linear Systems
- A.4. Algebraic Curves
- A.5. Abelian Varieties over C
- A.6. Jacobians over C
- A.7. Abelian Varieties over Arbitrary Fields
- A.8. Jacobians over Arbitrary Fields
- A.9. Schemes
- Part B. Height Functions
- B.1. Absolute Values
- B.2. Heights on Projective Space
- B.3. Heights on Varieties
- B.4. Canonical Height Functions
- B.5. Canonical Heights on Abelian Varieties
- B.6. Counting Rational Points on Varieties
- B.7. Heights and Polynomials
- B.8. Local Height Functions
- B.9. Canonical Local Heights on Abelian Varieties
- B.10. Introduction to Arakelov Theory
- Exercises
- Part C. Rational Points on Abelian Varieties
- C.1. The Weak Mordell-Weil Theorem
- C.2. The Kernel of Reduction Modulo p
- C.3 Appendix. Finiteness Theorems in Algebraic Number Theory
- C.4 Appendix. The Selmer and Tate-Shafarevich Groups
- C.5 Appendix. Galois Cohomology and Homogeneous Spaces
- Exercises
- Part D. Diophantine Approximation and Integral Points on Curves
- D.1. Two Elementary Results on Diophantine Approximation
- D.2. Roth's Theorem
- D.3. Preliminary Results
- D.4. Construction of the Auxiliary Polynomial
- D.5. The Index Is Large
- D.6. The Index Is Small (Roth's Lemma)
- D.7. Completion of the Proof of Roth's Theorem
- D.8. Application: The Unit Equation U + V = 1
- D.9. Application: Integer Points on Curves
- Exercises
- Part E. Rational Points on Curves of Genus at Least 2
- E.1. Vojta's Geometric Inequality and Faltings' Theorem
- E.2. Pinning Down Some Height Functions
- E.3. An Outline of the Proof of Vojta's Inequality
- E.4. An Upper Bound for h[subscript Omega](z, w)
- E.5. A Lower Bound for h[subscript Omega](z, w) for Nonvanishing Sections
- E.6. Constructing Sections of Small Height I: Applying Riemann-Roch
- E.7. Constructing Sections of Small Height II: Applying Siegel's Lemma
- E.8. Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version I
- E.9. Eisenstein's Estimate for the Derivatives of an Algebraic Function
- E.10. Lower Bound for h[subscript Omega](z, w) at Admissible (i*[subscript 1], i*[subscript 2]): Version II
- E.11. A Nonvanishing Derivative of Small Order
- E.12. Completion of the Proof of Vojta's Inequality
- Exercises
- Part F. Further Results and Open Problems
- F.1. Curves and Abelian Varieties
- F.1.1. Rational Points on Subvarieties of Abelian Varieties
- F.1.2. Application to Points of Bounded Degree on Curves
- F.2. Discreteness of Algebraic Points
- F.2.1. Bogomolov's Conjecture
- F.2.2. The Height of a Variety
- F.3. Height Bounds and Height Conjectures
- F.4. The Search for Effectivity
- F.4.1. Effective Computation of the Mordell-Weil Group A([kappa])
- F.4.2. Effective Computation of Rational Points on Curves
- F.4.3. Quantitative Bounds for Rational Points
- F.5. Geometry Governs Arithmetic
- F.5.1. Kodaira Dimension
- F.5.2. The Bombieri-Lang Conjecture
- F.5.3. Vojta's Conjecture
- F.5.4. Varieties Whose Rational Points Are Dense
- Exercises
- References
- List of Notation
- Index
- Part A. The Geometry of Curves and Abelian Varieties
- A.1. Algebraic Varieties
- A.1.1. Affine and Projective Varieties
- A.1.2. Algebraic Maps and Local Rings
- A.1.3. Dimension
- A.1.4. Tangent Spaces and Differentials
- A.2. Divisors
- A.2.1. Weil Divisors
- A.2.2. Cartier Divisors
- A.2.3. Intersection Numbers
- A.3. Linear Systems
- A.3.1. Linear Systems and Maps
- A.3.2. Ampleness and the Enriques-Severi-Zariski Lemma
- A.3.3. Line Bundles and Sheaves
- A.4. Algebraic Curves
- A.4.1. Birational Models of Curves
- A.4.2. Genus of a Curve and the Riemann-Roch Theorem
- A.4.3. Curves of Genus 0
- A.4.4. Curves of Genus 1
- A.4.5. Curves of Genus at Least 2
- A.4.6. Algebraic Surfaces
- A.5. Abelian Varieties over C
- A.5.1. Complex Tori
- A.5.2. Divisors, Theta Functions, and Riemann Forms
- A.5.3. Riemann-Roch for Abelian Varieties
- A.6. Jacobians over C
- A.6.1. Abelian Integrals
- A.6.2. Periods of Riemann Surfaces
- A.6.3. The Jacobian of a Riemann Surface
- A.6.4. Albanese Varieties
- A.7. Abelian Varieties over Arbitrary Fields
- A.7.1. Generalities
- A.7.2. Divisors and the Theorem of the Cube
- A.7.3. Dual Abelian Varieties and Poincare Divisors
- A.8. Jacobians over Arbitrary Fields
- A.8.1. Construction and Properties
- A.8.2. The Divisor [Theta]
- A.8.3 Appendix. Families of Subvarieties
- A.9. Schemes
- A.9.1. Varieties over Z
- A.9.2. Analogies Between Number Fields and Function Fields
- A.9.3. Minimal Model of a Curve
- A.9.4. Neron Model of an Abelian Variety