Qpoints - Rational points on varieties (notes from courses...

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Unformatted text preview: Rational points on varieties (notes from courses taught in Spring 2003 at Berkeley and Fall 2008 at MIT) Bjorn Poonen Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA E-mail address : poonen@math.mit.edu URL : http://math.mit.edu/~poonen/ 2000 Mathematics Subject Classication. Primary 14G05; Secondary 11G35 Key words and phrases. Rational points, descent, etale cohomology, torsors, Brauer-Manin obstruction, descent obstruction While writing these notes, the author was supported by NSF grants DMS-0301280 and DMS-0841321 and a Packard Fellowship. c Bjorn Poonen 2008. Contents Chapter 0. Introduction 1 0.1. Prerequisites 1 0.2. Standard notation 1 Chapter 1. Fields 3 1.1. Some elds arising in classical number theory 3 1.2. C r elds 4 1.3. Cohomological dimension 9 1.4. Brauer groups of elds 12 Exercises 22 Chapter 2. Varieties over arbitrary elds 25 2.1. Varieties 25 2.2. Base extension 26 2.3. Rational points 30 2.4. Closed points 37 Exercises 39 Chapter 3. Properties of morphisms 41 3.1. Finiteness conditions 41 3.2. Spreading out 43 3.3. Flat morphisms 48 3.4. Fppf and fpqc morphisms 50 3.5. Smooth and etale morphisms 51 3.6. Rational maps 65 3.7. Comparisons 68 Exercises 68 Chapter 4. Faithfully at descent 70 4.1. Motivation: glueing sheaves 70 4.2. Faithfully at descent for quasi-coherent sheaves 71 4.3. Faithfully at descent for schemes 72 4.4. Galois descent 74 iii 4.5. Twists 77 4.6. Restriction of scalars 81 Exercises 82 Chapter 5. Algebraic groups 84 5.1. Group schemes 84 5.2. Ane algebraic groups 89 5.3. Unipotent groups 90 5.4. Tori 92 5.5. Semisimple and reductive algebraic groups 93 5.6. Abelian varieties 94 5.7. Finite etale group schemes 96 5.8. Classication of algebraic groups 96 5.9. Torsors 97 Exercises 100 Chapter 6. Etale and fppf cohomology 102 6.1. The reasons for etale cohomology 102 6.2. Grothendieck (pre)topologies 103 6.3. Presheaves and sheaves 105 6.4. Cohomology 109 6.5. Torsors over an arbitrary base 113 6.6. Brauer groups 116 6.7. Spectral sequences 123 Exercises 127 Chapter 7. The Weil conjectures 129 7.1. Statements 129 7.2. The case of curves 130 7.3. Zeta functions 130 7.4. The Weil conjectures in terms of zeta functions 132 7.5. Cohomological explanation 133 7.6. Cycle class map 139 7.7. Applications to varieties over global elds 139 Exercises 141 Chapter 8. Cohomological obstructions to rational points 142 8.1. An example 142 8.2. Descent 145 8.3. The Brauer-Manin obstruction 153 iv 8.4. Comparing the descent and Brauer-Manin obstructions 157 8.5. Insuciency of the obstructions 160 Exercises 163 Chapter 9. Surfaces 164 9.1. Fano varieties 164 9.2. Kodaira dimension 165 9.3. Rational points on varieties of general type 167 9.4. Classication of surfaces 167 9.5. Del Pezzo surfaces 168 Exercises 170 Appendix A. Universes 171 1.1. Denition of universe 171 1.2. The universe axiom 171 1.3. Strongly inaccessible cardinals 1....
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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 University of Georgia Athens.

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Qpoints - Rational points on varieties (notes from courses...

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