Lectures on rational points on curves

Description

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