Book contents
- Frontmatter
- Contents
- Introduction
- 1 The Riemann zeta function
- 2 The zeta function of a Z-scheme of finite type
- 3 The Weil conjectures
- 4 L-functions from number theory
- 5 L-functions from geometry
- 6 Motives
- Appendix A Karoubian and monoidal categories
- Appendix B Triangulated categories, derived categories, and perfect complexes
- Appendix C List of exercises
- Bibliography
- Index
6 - Motives
Published online by Cambridge University Press: 28 April 2020
- Frontmatter
- Contents
- Introduction
- 1 The Riemann zeta function
- 2 The zeta function of a Z-scheme of finite type
- 3 The Weil conjectures
- 4 L-functions from number theory
- 5 L-functions from geometry
- 6 Motives
- Appendix A Karoubian and monoidal categories
- Appendix B Triangulated categories, derived categories, and perfect complexes
- Appendix C List of exercises
- Bibliography
- Index
Summary
This last chapter is dedicated to motives and their zeta functions. We consider only an elementary case: that of pure motives of Grothendieck associated to smooth projective varieties over a finite field. One can go much further using the triangulated categories of motives introduced by Voevodsky and developed by Ivorra, Ayoub, and Cisinski–D’eglise, but this would go beyond the scope of the book. Nonetheless, it is explained how this viewpoint considerably clarifies how Weil cohomologies are used to prove rationality and the functional equation, and this theory is applied to prove a somewhat forgotten theorem of Weil: Artin’s conjecture for non-abelian L-functions in positive characteristic.
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- Zeta and L-Functions of Varieties and Motives , pp. 142 - 167Publisher: Cambridge University PressPrint publication year: 2020