Lectures on rational points on curves

✍ Scribed by Bjorn Poonen

• Year: 2006
• Tongue: English
• Leaves: 118
• Category: Library

✦ Table of Contents

Chapter 0. Introduction
0.1. Notation
Chapter 1. Varieties over perfect fields
1.1. Affine varieties
1.2. Projective varieties
1.3. Base extension
1.4. Irreducibility
1.5. Morphisms and rational maps
1.6. Dimension
1.7. Smooth varieties
1.8. Valuations and ramification
1.9. Divisor groups and Picard groups
1.10. Twists
1.11. Group varieties
1.12. Torsors
Exercises
Chapter 2. Curves
2.1. Smooth projective models
2.2. Divisor groups and Picard groups of curves
2.3. Differentials
2.4. The Riemann-Roch theorem
2.5. The Hurwitz formula
2.6. The analogy between number fields and function fields
2.7. Genus-0 curves
2.8. Hyperelliptic curves
2.9. Genus formulas
2.10. The moduli space of curves
2.11. Describing all curves of low genus
Exercises
Chapter 3. The Weil conjectures
3.1. Some examples
3.2. The Weil conjectures
3.3. The case of curves
3.4. Zeta functions
3.5. The Weil conjectures in terms of zeta functions
3.6. Characteristic polynomials
3.7. Computing the zeta function of a curve
Exercises
Chapter 4. Abelian varieties
4.1. Abelian varieties over arbitrary fields
4.2. Abelian varieties over finite fields
4.3. Abelian varieties over C
4.4. Abelian varieties over finite extensions of Qp
4.5. Cohomology of the Kummer sequence for an abelian variety
4.6. Abelian varieties over number fields
Exercises
Chapter 5. Jacobian varieties
5.1. The Picard functor and the definition of the Jacobian
5.2. Basic properties of the Jacobian
5.3. The Jacobian as Albanese variety
5.4. Jacobians over finite fields
5.5. Jacobians over C
Exercises
Chapter 6. 2-descent on hyperelliptic Jacobians
6.1. 2-torsion of hyperelliptic Jacobians
6.2. Galois cohomology of J[2]
6.3. The x-T map
6.4. The 2-Selmer group
Exercises
Chapter 7. Étale covers and general descent
7.1. Definition of étale
7.2. Constructions of étale covers
7.3. Galois étale covers
7.4. Descent using Galois étale covers: an example
7.5. Descent using Galois étale covers: general theory
7.6. The Chevalley-Weil theorem
Exercises
Chapter 8. The method of Chabauty and Coleman
Chapter 9. The Mordell-Weil sieve
Acknowledgements
Bibliography