curves - Lectures on rational points on curves March 5 2006...

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Unformatted text preview: Lectures on rational points on curves March 5, 2006 version Bjorn Poonen Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail address : [email protected] URL : http://math.berkeley.edu/~poonen This research was supported by NSF grant DMS-0301280 and the Miller Institute for Basic Research in Science. c Bjorn Poonen 2006. Contents Chapter 0. Introduction 1 0.1. Notation 1 Chapter 1. Varieties over perfect fields 3 1.1. Affine varieties 3 1.2. Projective varieties 5 1.3. Base extension 7 1.4. Irreducibility 7 1.5. Morphisms and rational maps 8 1.6. Dimension 10 1.7. Smooth varieties 11 1.8. Valuations and ramification 13 1.9. Divisor groups and Picard groups 14 1.10. Twists 16 1.11. Group varieties 17 1.12. Torsors 19 Exercises 21 Chapter 2. Curves 23 2.1. Smooth projective models 23 2.2. Divisor groups and Picard groups of curves 24 2.3. Differentials 25 2.4. The Riemann-Roch theorem 26 2.5. The Hurwitz formula 28 2.6. The analogy between number fields and function fields 31 2.7. Genus-0 curves 35 2.8. Hyperelliptic curves 36 2.9. Genus formulas 39 2.10. The moduli space of curves 41 2.11. Describing all curves of low genus 43 iii Exercises 45 Chapter 3. The Weil conjectures 49 3.1. Some examples 49 3.2. The Weil conjectures 50 3.3. The case of curves 51 3.4. Zeta functions 52 3.5. The Weil conjectures in terms of zeta functions 53 3.6. Characteristic polynomials 54 3.7. Computing the zeta function of a curve 54 Exercises 55 Chapter 4. Abelian varieties 57 4.1. Abelian varieties over arbitrary fields 57 4.2. Abelian varieties over finite fields 62 4.3. Abelian varieties over C 68 4.4. Abelian varieties over finite extensions of Q p 70 4.5. Cohomology of the Kummer sequence for an abelian variety 72 4.6. Abelian varieties over number fields 73 Exercises 76 Chapter 5. Jacobian varieties 79 5.1. The Picard functor and the definition of the Jacobian 79 5.2. Basic properties of the Jacobian 80 5.3. The Jacobian as Albanese variety 81 5.4. Jacobians over finite fields 82 5.5. Jacobians over C 83 Exercises 84 Chapter 6. 2-descent on hyperelliptic Jacobians 87 6.1. 2-torsion of hyperelliptic Jacobians 87 6.2. Galois cohomology of J [2] 90 6.3. The x- T map 91 6.4. The 2-Selmer group 92 Exercises 92 Chapter 7. ´ Etale covers and general descent 93 7.1. Definition of ´ etale 93 iv 7.2. Constructions of ´ etale covers 93 7.3. Galois ´ etale covers 97 7.4. Descent using Galois ´ etale covers: an example 98 7.5. Descent using Galois ´ etale covers: general theory 101 7.6. The Chevalley-Weil theorem 102 Exercises 102 Chapter 8. The method of Chabauty and Coleman 105 Chapter 9. The Mordell-Weil sieve 107 Acknowledgements 109 Bibliography 111 v CHAPTER 0 Introduction The main goal of these notes is to explain how to answer questions like “What are the rational number solutions to x 4 + y 4 = 17?” Warning 0.0.1 . These notes are not yet finished, and what is written is only a first draft. Read them at your own risk! If you seedraft....
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This note was uploaded on 10/26/2011 for the course MATH 8320 taught by Professor Clark during the Fall '10 term at UGA.

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curves - Lectures on rational points on curves March 5 2006...

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