Navigating the motivic world

Contents Introduction 1 Chapter 1. Introduction to the Weil conjectures 3 1. A first look 3 2. Formal statement of the conjectures 8 3. Zeta functions 11 4. A plan to prove the conjectures 14 5. Some history of the proofs of the conjectures 18 A. Computer calculations 20 B. Computations for diagonal hypersurfaces 25 Chapter 2. Topological interlude: the cohomology of algebraic varieties 35 1. Lefschetz theory 35 2. The Hard Lefschetz theorem 37 3. The Hodge index theorem 38 4. Hodge theory 40 5. Correspondences and the Lefschetz fixed point theorem 43 Chapter 3. A second look at the Weil conjectures 49 Chapter 4. Introduction tó etale cohomology 51 1. Overview of some key points 51 2. Topological perspectives 53 3. Rigid open covers and generalizedČech complexes 58 4. Cohomology viá etale coverings 64 5. ´ Etale maps in algebraic geometry 70 6. Systems of approximations 71 7. Hypercovers andétale homotopy types 73 8. ´ Etale cohomology andétale K-theory 75