Diophantine Geometry

✍ Scribed by Marc Hindry, Joseph H. Silverman

• Publisher: Springer
• Year: 2000
• Tongue: English
• Leaves: 570
• Category: Library

✦ Table of Contents

Cover
Title Page
Copyright
Dedication
Preface
Contents
Acknowledgments
Contents
Detailed Contents for Part A
Introduction
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 Poincar6 Divisors
A.8 Jacobians over Arbitrary Fields
A.8.1 Construction and Properties
A.8.2 The Divisor O
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 N6ron Model of an Abelian Variety
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 hn(z, w)
E.5 A Lower Bound for hn(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 ho (z, w) at Admissible (ii, i2): Version I
E.9 Eisenstein's Estimate for the Derivatives of an Algebraic Function
E.10 Lower Bound for ho (z, w) at Admissible (ii, i2): 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(k)
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
Back Cover